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Mathematical Methods in Undergraduate Physics-Mechanics

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Mathematical Methods in Undergraduate Physics-Mechanics Powered By Docstoc
					              Educational Systems®
              Knowledge Engineering Foundation
              Assessment and Learning Institute
              37 Κoliatsou Street,
              Corinthos 20100, Greece
              solomon_antoniou@yahoo.com




            Mathematical Methods
                       in Physics

             (Symbolic and Numerical Methods
             in Undergraduate Physics Courses)


                         Mechanics
              One-Dimensional Rectilinear Motion



                 Classification of Theory-
                -Methods and Techniques
                   Dr. Solomon M. Antoniou


Copyright© by Educational Systems®
         ©
Philosophy


This course tries to combine
         • Traditional mathematical methods
with
         • Symbolic and Numerical techniques


For this purpose there is
          • A short exposition of theory
          • Emphasis on mathematical techniques
          • Classification of methods and techniques
          • Complete step-by-step solutions and proofs




                                                         2
Objectives


The objectives of this Chapter is the student to be able
         • To solve problems in one-dimensional Rectilinear Motion
             • Horizontal or
             • Vertical
          using ordinary differential equations
          • To develop code in any of the symbolic languages
             in order to solve the differential equations of motion
          • To develop code in any of the symbolic languages
             in order to plot the solutions of the equations of motion




                                                                         3
Contents
                                Chapter 1
               One-Dimensional Rectilinear Motion.


1.0.   Symbolic Solutions of Ordinary Differential Equations
1.1.   Horizontal Motion
              1.1.1. First Case. Constant Force
              1.1.2. Second Case. Force depends on the instantaneous
                    velocity.
              1.1.3. Third Case. Force depends on time.
              1.1.4. Fourth Case. Force depends on replacement
1.2.   Vertical Motion
              1.2.1. Vertical Upward
              1.2.2. Vertical Downward
              1.2.3. Motion in Earth’s Gravitational Field
              1.2.4. Motion of a chain
1.3.   Work
              1.3.1 Work-Energy Theorem
              1.3.2 Conservative Forces
1.4.   Momentum
              1.4.1 Momentum Conservation
1.5.   Impulse
              1.5.1 Impulse-Momentum Theorem




                                                                       4
Learning Outcomes


By the time the student completes this chapter, he/she will know
          • How to write down Newton’s differential equation
                             d2 x              dv
                         m          = F or m      =F
                             dt 2              dt
            in one of the following cases


            A. Horizontal Motion
            1. When the force is constant
               1.1.   the force points the direction of motion
               1.2.   the force opposes the motion
            2. When the force depends on the instantaneous velocity
            3. When the force depends on time
            4. When the force depends on the displacement


            B. Vertical Motion
            1. Upward Vertical Motion
            2. Downward Vertical Motion
            3. Motion in Earth’s Gravitational Field
            4. Motion of a Chain




                                                                      5
• How to solve the differential equation established in one of
   the two previous cases and determine
       1. The displacement
       2. The velocity
       3. The acceleration
       4. The time taken
  under some initial conditions imposed as mathematical
  relations based on the physical principles of the problem
  and using any one of the two methods
  a. finding the general solution of the differential
     equation and determining the arbitrary constant(s)
     taking into account the initial conditions or
  b. integrating the differential equation and using
   the initial conditions as limits in the definite integrals (since the differential
   equation is, in most of the cases, a first order differential equation).



• How to develop code for the solution of any of the
   differential equations, using any of the Symbolic Languages
   (Macsyma, Maple, Axiom, Mathematica, Reduce, Derive, Matlab,
   Maxima, MuPad and Scientific WorkPlace).



• How to develop code for plotting the solution of the
   equations of motion.




                                                                                   6
                                    Chapter 1
              One-Dimensional Rectilinear Motion.


1.0. Symbolic Solutions of Ordinary Differential Equations.

1.0.1. Macsyma (Version 2.4.0)
Macsyma is the best available package (in the sense of user-friendly and reliability) in
the market.
On the other hand you may find a lot of add-on packages available, performing a
number of very complicated tasks (like Painlevé analysis, Non-linear Partial
Differential equations, exploiting Lie Symmetries in solving differential equations,
etc).
Example 1.
                                                     3
                                        d2y      dy 
Step 1. Enter the differential equation    2
                                             + y  = 0 by assigning the operator
                                        dx       dx 
eqn to the differential equation.




Step2. Find the general solution of the equation using the command ode, where
gensol is the operator corresponding to the general solution




where %k1 and %k 2 are constants.




                                                                                       7
                                                dy
Step 3. Enter the initial condition y(0) = 0,                = 2 and get the answer
                                                dx   (0,0)




Step 4. Enter the boundary condition y(0) = 1, y(1) = 3 and get the answer




Example 2. The system
                                  dx         
                                     =y      
                                  dt         
                                             
                                 dy          
                               m    = −k y 2 
                                 dt          
can be solved under the conditions x (0) = 0, y(0) = p using the code
References:
   1. Macsyma User’s Guide, A Tutorial Introduction
       Second Edition 1996 (pp.95-107)

1.0.2. Scientific WorkPlace (Version 2.5)
From the menu choose Maple and then
                 Solve ODE/Exact, Laplace, Numeric or Series.
Example 1.
   •   Step 1.
       Enter (change from text T mode in mathematics M mode) the expression



   •   Step 2.
       From the menu bar choose Maple and then Solve ODE/Exact
   •   Step 3.
       In the appearing ODE Independent Variable box enter the independent
       variable x and press OK. The program responds with




                                                                                      8
Example 2.
   •   Step 1.
       From the menu choose Insert and then Display.
   •   Step 2.
       Enter within the frame the following (pressing Enter after each one of the
       equations)




       The reader may easily understand that the first equation is a second order
       differential equation. The last two equations are the initial conditions.
   •   Step 3.
       From the menu bar choose Maple and then Solve ODE/Exact
   •   Step 4.
       In the appearing ODE Independent Variable box enter the independent
       variable x and press OK. The program responds with



Example 3. The system
                                  dx         
                                     =y      
                                  dt         
                                             
                                 dy          
                               m    = −k y 2 
                                 dt          
can be solved under the conditions x (0) = 0, y(0) = p using the code
References:
   1. Doing Mathematics with Scientific WorkPlace™,
       Darel W. Hardy and Carol L. Walker (pp.233-249)
       Brookes/Cole 1995




                                                                                    9
   1.03. Maple (Version 7.00)
Example 1.
                                                   dy
Step 1. We introduce the differential equation x      + y = x2
                                                   dx




Step 2. We use the command dsolve in order to solve the differential equation




Step 3. We use initial condition




Step 4. We get a special solution of the differential equation by determining the
arbitrary constant.




Example 2. The system
                                   dx     
                                      =y  
                                   dt     
                                          
                                dy       2
                              m    = −k y 
                                dt        
can be solved under the conditions x (0) = 0, y(0) = p using the code




                                                                                    10
References:
   1. Maple V Language Reference Manual
       Bruce W. Char et al., Springer-Verlag 1991
       (pp.23-24)



1.0.4. Reduce (Version 3.6)
This very popular package written by Anthony C. Hearn is powerful enough and
has been used by the scientific community in so many diverse fields (Mathematics,
Astronomy, High Energy Physics, etc). To the best of my knowledge, this package
can only solve first order differential equations using the user-contributed package
ODESOLVE.
Step1. Load the package ODESOLVE




Step 2. The following command depend says that y is a function of x.



                                                   dy
Step 3. We introduce the differential equation x      + y = x2
                                                   dx




                                                                                       11
Step 4. We use the command odesolve in order to solve the differential equation



and then we get the answer




References:
    1. Reduce 3.6
        Codemist Ltd 1995
        Chapter 54, (pp.345-346)
1.0.5. Axiom (Version 2.0a, NAG, Sun SparcStation 4)
                                                         d2y       dy
In order to solve the differential equation ( x 2 + 1)      2
                                                              + 3x    + y( x ) = 0 ,
                                                         dx        dx
we enter
                deq : = (x**2+1)* D(y x, x, 2)+3*x*D(y x, x)+y x=0
and then use the command
                solve(deq, y, x)
We get the answer
                            x 3  ( − x / 2)                     x 3   
[particular = 0 , basis = cos   %e
                               2             , %e ( − x / 2) sin    
                                                                    2    ]
                           
                                                                       
                                                                           
in a self-explanatory notation.
Example 2. The system
                                     dx         
                                        =y      
                                     dt         
                                                
                                    dy          
                                  m    = −k y 2 
                                    dt          
can be solved under the conditions x (0) = 0, y(0) = p using the code
References:
    1. Axiom, The Scientific Computation System
        Richard D. Jenks and Robert S. Sutor
        Springer-Verlag 1992, (NAG, Oxford)




                                                                                       12
1.06. Matlab (Version 4)


Example 1.
                                                          dy
Step1. In order to solve the differential equation           = 1 + y 2 with initial condition
                                                          dx
y(0) = 1 , enter



Step 2. You get the result




Example 2.
                                                          d2y
Step1. In order to solve the differential equation             + y = 4 cos( t ) with initial
                                                          dt 2
            π       dy
conditions y  = 2 ,                   = −3 , enter
            2       dt     ( π / 2)




Step 2. You get the result




Example 3. The system
                                           dx         
                                              =y      
                                           dt         
                                                      
                                          dy          
                                        m    = −k y 2 
                                          dt          
can be solved under the conditions x (0) = 0, y(0) = p using the code
References:
   1. The student edition of MATLAB, Version 4
       User’s Guide , Prentice Hall 1995




                                                                                                13
1.0.7. Mathematica (Version 3.0)
Symbolic solution of an Ordinary Differential Equation can be implemented using the
command DSolve[ ].


Example 1. The code for the solution of the differential equation
                                y ′( x ) = k ⋅ y( x )
is


The response you get from Mathematica (Shift +Enter) is



where C[1] is, as you can easily recognise, an arbitrary constant.
In order to determine the constant C[1], we define the function (copy-paste from
previous output)




and then solve the equation




which determines the value of the constant C[1] to be just 1, using the condition
y ( 0) = 1 .
         If you include boundary conditions, for example y(0) = 1 , in the code, the
constant C[1] appearing in the general solution will take a specific value.
In this case the command is



and you get the response


                                           dy
Example 2. The differential equation          = k − x has a solution subject to the
                                           dx
condition y(0) = 0 which can be found using the code



                                                                                       14
Example 3. The differential equation y ′′ + 2 y ′ + y = 0 has a solution subject to the
conditions y(0) = 1, y ′(0) = −2 which can be found using the code




Example 4. The differential equation y ′′ − 2 y ′ + y = x 2 e x has a solution subject to the
conditions y(0) = 1, y ′(0) = 0 which can be found using the code




Example 5. The system
                                    dx        
                                       =y     
                                    dt        
                                              
                                  dy          
                                m    = −k y 2 
                                  dt          
can be solved under the conditions x (0) = 0, y(0) = p using the code




References:
    1. Mathematica, A system for doing mathematics by computer
        S. Wolfram, Third Edition
    2. A Physicist’s guide to MATHEMATICA
        Patrick T. Tam, AP 1997
    3. Mathematica by Example
        Martha L. Abell, James P. Braselton
        AP 1997



                                                                                          15
1.0.8. Derive (for Windows)
Step 1. From the menu Author, enter y( x ) : = and press Enter. You get the response



Step 2. From the menu Author, enter x*dif(y(x),x)+y(x) = x^2 and press Enter.
You get the response




Step 3. From the menu Calculus, enter Integrate and press Enter. You get




Step 4. From the menu press Simplify and Enter. You get




Step 4. From the menu Author press F3 and then a. You get the general solution of
the differential equation #2




References:
   1. Improving Mathematics Teaching with DERIVE,
       Bernhard Kutzler, Chartwell-Bratt 1996
       (pp.93-98)
   2. Introduction to DERIVE for Windows,
       B. Kutzler, 1996




                                                                                       16
1.1. Horizontal Motion.
       In this case the motion of the particle is being conducted in a horizontal axis
(which might be either horizontal or vertical).




In the case of Horizontal Motion, we consider previous Figure 1.
In this case we have
       O is the origin of the motion.
       r
       i is the unit vector.

           OA is the vector displacement
           r
           v is the instantaneous velocity of the particle at a given point A
We usually put

                                  OA = x = x i                                   (1)
where x is the displacement, which is considered to be a function of time:
                                  x = x (t )                                    (2)
             r
The velocity v at any time t, is the first derivative of the displacement with respect
to time:
                                      r
                                  r dx
                                  v=                                    (3)
                                     dt
The speed v is the magnitude of the velocity (the component of the velocity vector)
defined by
                                  r      r
                                  v = v⋅ i                                       (4)

                 r                                                              r
The acceleration a at time t is defined as the first derivative of the velocity v :
                                 r
                           r dv
                           a=                                              (5)
                                dt




                                                                                         17
It is therefore obvious that the acceleration is also the second derivative of the
displacement with respect to time:
                                    2r
                                 r d x
                                 a= 2                                   (6)
                                   dt
The basic differential equation which governs the motion is Newton’s Second Law:
                                  r
                               d2x r
                             m 2 =F                            (7)
                               dt
or
                                    r
                                   dv r
                                 m    =F                                (8)
                                   dt
      r
where F is the total force (resulting force) acting on the particle.
It is much easier to solve the differential equation (8) instead of (7), because it is a
first order differential equation.
Equation (8) can be integrated very easily. It is obvious that the general solution will
contain an arbitrary constant, which can be determined using the initial conditions.
The initial conditions can be taken into account either
     • by finding the general solution and then determining the arbitrary constant
or
     • by using definite integrals so as the initial conditions appear as limits of the
        integrals.


Remarks:
1.       The origin of the axis is always the starting point of the motion.
                                                                          r
2.       The positive direction of the axis (direction of the unit vector i ) is the
         direction of the motion of the particle.




                                                                                           18
Horizontal Motion --- Solution Strategy
Stage 1.
      Draw a diagram consisting of
      1. A horizontal axis
      2. A point O as an origin
      3. Put an arrow (parallel to the axis and in some reasonable
           distance from it) showing the direction of motion of the particle

      4. Put a unit vector i having as origin the origin O of the axis
           and direction the positive direction (the direction of the motion)
      5. Consider an arbitrary position A on the axis
           Consider at this point A

              a. The velocity vector v
                  pointing to the direction of the unit vector
              b. The resultant Force F
Stage 2.
      Consider the differential equation of the motion (Newton’s Second
            d2 x
      Law) m 2 = F
            dt
      1. Split the differential equation into the system

                                   dx     
                                       =v 
                                   dt     
                                           (S1)
                                    dv    
                                  m = F
                                    dt    
      2. The above system, since

                     x = x ⋅i , v = v⋅i , F = F⋅i
           may be written in equivalent form as



                                                                           19
                                       dx     
                                           =v 
                                       dt     
                                               (S2)
                                        dv    
                                      m = F
                                        dt    
       3. Solve the system (S2) using EITHER
           a. Symbolic Algebra packages (Macsyma, Mathematica, Maple,
             Scientific WorkPlace, Matlab, Derive, Reduce, Maxima,
             MuPad Pro, Axiom, etc) OR
                                                                  dv
           b. Solve first the 1st order differential equation m      = F in
                                                                  dt
             each one of the cases of Appendix A (and then
                                       dx
             differential equation        = v , or a combination between them)
                                       dt

Stage 3.
Plot the various equations – using Symbolic Algebra Packages
       1. Displacement vs. Time
       2. Velocity vs. Time
       3. Acceleration vs. Time
       4. Displacement vs. Velocity
       5. Acceleration vs. Displacement


How to Consider Initial (or Boundary) Conditions.
(How to translate physical conditions into mathematical relations)
   1. If the initial displacement is x 0 this means that for t = 0 , x = 0 .
       In other words, x (0) = x 0 .
   2. If there is not any initial displacement, then x (0) = 0
   3. If the initial velocity is v 0 this means that for t = 0 , v = 0 .
       In other words, v(0) = v 0 .



                                                                              20
   4. If there is not any initial velocity, then v(0) = 0 .
   5. If the particle comes to rest after travelling some distance, this
      means that the final velocity is zero. In other words, v( t ) = 0


How to implement the Initial Conditions
          I)     By determining the constants in the general solutions of
                 the differential equations
          II)    By considering the limits of the integrals (integrating the
                 differential equations)
Appendix A.            Solving the system
                                     dx     
                                         =v 
                                     dt     
                                             (S2)
                                      dv    
                                    m = F
                                      dt    
in various cases of the force F.
First Case. Constant Force.
If F = const. , then
                                 dv
                             m      =F
                                 dt
is a separable differential equation and may be integrated,
                             m ∫ dv = F∫ dt                        (1)

giving the result
                             mv = F⋅ t + C                         (2)
where C is an arbitrary constant, determined from the initial conditions.
                                   dx
The first differential equation       = v of the system (S2), can be written,
                                   dt
because of (2) in the form




                                                                           21
                                  dx
                              m      = F⋅t + C                               (3)
                                  dt
which is a separable differential equation and may be integrated,
                              m ∫ dx = ∫ (F ⋅ t + C) dt                      (4)

giving the result
                             1
                      m ⋅ x = F ⋅ t 2 + C ⋅ t + C1                           (5)
                             2
where C1 is another constant of integration, determined from the initial
conditions.
If the initial velocity is v 0 , (for t = 0, v = v0 ), then we get from (2):
m v 0 = F ⋅ 0 + C and then

                              C = m v0                                       (6)
Eq.(2) gives us m v = F ⋅ t + m v 0 which is equivalent to
                                      F
                              v=        ⋅ t + v0                                    (7)
                                      m
If the initial displacement is zero (i.e. for t = 0, x = 0 ) then
                         1
we get from (5) m ⋅ 0 = F ⋅ 0 + C ⋅ 0 + C1 and so C1 = 0 .
                         2
                          1
Eq.(5) becomes m ⋅ x = F ⋅ t 2 + mv0 ⋅ t which is equivalent to
                          2
                                      1 F 2
                              x=         ⋅ t + v0 ⋅ t                        (8)
                                      2m
Using Definite Integrals
If the initial velocity is v 0 , (for t = 0, v = v0 ), then we get from (1)
                                  v          t
                              m ∫ dv = F ∫ dt
                                v0           0

which is equivalent to

                  m[v ]v 0 = F[t ]0 ⇔ m( v − v 0 ) = F( t − 0) ⇔ v − v 0 =
                                                                             F
                       v
                                  t
                                                                               ⋅t
                                                                             m


                                                                                     22
or
                                     F
                                v=     ⋅ t + v0
                                     m
which is eq.(7).
If the initial displacement is zero (i.e. for t = 0, x = 0 ) then we get from (4)
                x          t
              m ∫ dx = ∫ (F ⋅ t + C)dt
                  0        0

which is equivalent to
                                          t
                             t2                     t2
              m[x ]   x
                      0   = F ⋅ + C ⋅ t  or mx = F ⋅ + C ⋅ t
                             2
                                        0
                                                     2

from which we get
                                     1 F 2
                                x=      ⋅ t + v0 ⋅ t
                                     2m
which is eq.(8).
Second Case. Force depends on instantaneous velocity.
                           dv
If F = F( v) then m           = F may be written as
                           dt
                      dv
               m         = F( v)                                 (1)
                      dt
which is a separable differential equation and may be integrated giving
                       dv                      dv
              m∫           = dt    or m ∫           =t+C
                      F( v) ∫
                                                                       (2)
                                              F( v)
where C is an arbitrary constant determined from the initial conditions.
Equation (2) expresses time in terms of the velocity.
                                          dv
The same differential equation m             = F may be written as
                                          dt
                  dv dx             dv
              m         = F( v) or m v = F( v)                         (3)
                  dx dt             dx
which is a separable differential equation and may be integrated, giving


                                                                              23
                      v                      v
                m∫         dv = ∫ dx or m ∫       dv = x + C1               (4)
                     F( v)                  F( v)
C1 is a constant determined from the initial conditions.
Equation (4) expresses displacement in terms of velocity.
Using Definite integrals
If the initial velocity is v 0 (i.e. for t = 0, v = v 0 ) we get from (1) after
separation of variables and integration
                       v               t
                             dv
                     m∫          = dt
                       v0
                            F( v) ∫
                                  0

which is equivalent to
                                  v
                                        dv
                         t=m∫                                 (5)
                                  v0
                                       F( v)

If the initial displacement is zero (i.e. for t = 0, x = 0 ), we get from (3)
                                                         v             x
                                                              v
after separation of variables and integration m ∫                  dv = ∫ dx , which is
                                                        v0
                                                             F( v)      0

equivalent to
                                   v
                                        v
                         x=m∫                dv               (6)
                                  v0
                                       F( v)

Third Case. Force depends on time.
                                                  dv
If F = F( t ) the differential equation m            = F may be written as
                                                  dt
                    dv
                m      = F( t )                                             (1)
                    dt
which is a separable differential equation and may be integrated, giving
m ∫ dv = ∫ F( t )dt or

                m ⋅ v + C = ∫ F( t )dt                                      (2)

where C is a constant determined from the initial conditions.


                                                                                     24
Equation (2) expresses the velocity in terms of time.
The other differential equation
                               dx
                                  =v                                   (3)
                               dt
is also separable and may be integrated, giving ∫ dx = ∫ v dt or

                               x + C1 = ∫ v dt                         (4)

where v is a function of time, as follows from (2).
Equation (2) expresses displacement x in terms of time t .
Using Definite integrals
If the initial velocity is v 0 (i.e. for t = 0, v = v0 ) we get from (1) after
separation of variables and integration
                                    v            t
                               m ∫ dv = ∫ F( t )dt
                                    v0           0

which is equivalent to
                                                     t
                               m( v − v0 ) = ∫ F( t )dt
                                                     0

from which we get
                                             t
                                    1
                         v = v0 +
                                    m∫
                                        F( t )dt                (5)
                                      0

If the initial displacement is zero (i.e. for t = 0, x = 0 ),
we get from (3) after separation of variables and integration
x       t
∫ dx = ∫ v( t )dt   which is equivalent to
0       0

                                         t
                               x = ∫ v( t )dt                          (6)
                                         0

and v( t ) is given by (5).
Second Method. This method refers to § 1.1.3.


                                                                                 25
Fourth Case. Force depends on displacement
                         dv
If F = F( x ) then m        = F may be written as
                         dt
                   dv              dv dx             dv
               m      = F( x ) ⇔ m       = F( x ) ⇔ m v = F( x ) (1)
                   dt              dx dt             dx
which is a separable differential equation and may be integrated giving
m ∫ v dv = ∫ F( x ) dx

or
                              1
                                mv 2 + C = ∫ F( x ) dx                  (2)
                              2
where C is a constant determined from the initial conditions.
Equation (2) expresses a relation between the velocity v and the
displacement x of the particle.
Solving eq.(2) with respect to v we may integrate the first differential
equation
                              dx
                                 =v                                     (3)
                              dt
          dx
giving   ∫ v = ∫ dt or
                                dx
                             ∫ v( x) = t + C1                           (4),

where v( x ) comes from (1).
Equation (2) expresses the time t in terms of the displacement x.
Using Definite integrals
                                             dv
If the initial velocity is v 0 (i.e. for m      = F ) and the initial
                                             dt
displacement is x = 0 , we get from (1) after separation of variables and
integration




                                                                               26
                                v              x
                              m ∫ v dv = ∫ F( x ) dx
                                v0             0

                                                   x
                          1
which is equivalent to      m( v 2 − v0 ) = ∫ F( x )dx from which we get
                                      2
                          2                 0

                                                   x
                                               2
                              v 2 = v0 +
                                     2
                                               m∫
                                                   F( x )dx                    (5)
                                                 0

If the initial displacement is zero (i.e. for t = 0, x = 0 ), we get from (3)
                                                          x       t
                                                             dx
after separation of variables and integration             ∫ v(x ) = ∫ dt which is
                                                          0         0


equivalent to
                                 x
                                       dx
                              t=∫                                              (6)
                                 0
                                      v( x )

and v( t ) is given by (5).
Second Method. We solve the second order differential equation
                                     d2x
                               m          = F( x )
                                     dt 2
Third Method. This method refers to § 1.1.4.




                                                                                     27
                                            r
1.1.1 First Case. Constant Force, F = const.
Example 1. A particle of mass m moves along a straight line under the influence
of a constant force of magnitude F (Fig.2).




We suppose that the particle has an initial speed v 0 . Find
(i) the velocity at any time t.
(ii) the displacement x as a function of a) the speed b) of the time.
(iii) the speed at any time t.



Solution.
(i)      Using Newton’s second law, we have
                          r
                         dv r         dv r     r
                      m     =F⇔ m         i = Fi ⇔
                         dt            dt
                                          dv
                                    ⇔ m      =F                             (1)
                                          dt
Indefinite Integrals.
Separating the variables, we get m dv = F dt from the previous equation
and after integration,

                                  m∫ dv = F∫ dt                             (2)

we get
                                  mv = F ⋅ t + C                    (3)


where C is a constant. We have now to use the initial conditions in order
to determine constant C.




                                                                                  28
Since for t = 0 the speed is v = v 0 , we get from (3):
                                       mv 0 = F ⋅ 0 + C
from which we get
                                    C = mv 0                                  (4)
and so eq.(3) becomes: mv = F ⋅ t + mv 0             which is equivalent to
                                              F
                                 v = v0 +       ⋅t                                  (5)
                                              m
Definite Integrals.
We could get the same result, eq.(5) ,if instead of using indefinite integrals in eq.(2),
we would have been using definite integrals instead. In this case, taking into account
the initial conditions, eq.(2) gives us
                                   v           t

                                 m ∫ dv = F ∫ dt                                    (6)
                                   v0          t0


where there is an obvious correspondence between the limits of integration:
                t = 0→v = v 0          and          t=t → v=v

Performing the integration, we get

                                 m[v ]v = v 0 = F[t ]0 ⇔
                                        v=v          t



                                 m( v − v 0 ) = F t ⇔
                                              F
                                 v − v0 =       t
                                              m
which is further equivalent to
                                              F
                                 v = v0 +       ⋅t
                                              m
which is again eq. (5).




        Using Mathematica, the code for deriving eq. (5) is




and after pressing Shift+Enter you get the answer


                                                                                          29
                  r
(ii) The velocity v is given obviously by
                        r         F r
                        v =  v 0 + t i                                     (7)
                                  m 
(iii) For the calculation of the displacement x , instead of using the second
                                  r
                               d2x r
order differential equation   m 2 =F                 we can use the first-order equation
                               dt
    r
   dv r
 m    =F     and then the obvious relation
   dt
                         dv dv dx dv
                           =     =   v                                       (8)
                         dt dx dt dx
in order to eliminate the time t .
                                                        dv
So, starting from the differential equation m              = F of the previous section,
                                                        dt
we get
                                     dv dx
                                 m         =F           ⇔
                                     dx dt
                                     dv
                                 m      v=F                                  (9)
                                     dx
The previous equation can be solved by separation of variables and integration:
                                 m v dv = F dx ,
                                     v           x

                                 m ∫ v dv = F∫ dx ,
                                     v0          0

                                          v=v
                                   v2 
                                               = F [ x] 0 ,
                                                        x
                                 m 
                                   2  v = v0

                                   v2 v2 
                                 m − 0  = F x                              (10)
                                   2  2

from which we get
                                                2F
                                 v2 − v2 =
                                       0           x
                                                m
which is further equivalent to




                                                                                           30
                                              F
                              v2 = v2 + 2
                                    0           x                         (11)
                                              m
and

                               x=
                                    m 2
                                    2F
                                       ( v − v 20 )                       (12)

The previous equation expresses the displacement x as a function of the speed.




       Using Mathematica, the code for deriving eq. (12) is




and after pressing Shift+Enter you get the answer




Using the value of v from eq.(5) into eq.(12), we get

                            m  F              
                                           2
                                         
                       x=        t + v0  − v2  ,
                               
                            2F  m           0
                                                

                          m  F2 2    F     
                       x=     2 t + 2 t v0  ,
                          2F  m      m     

                            1 F 2
                       x=      t + t v0
                            2m
which can be written as
                                   1F 2
                      x = v0 t +      t                        (13)
                                   2m
The above equation expresses the displacement x as a function of time.
2nd Method of deriving eq.(13)
                                                         dx
We could get eq.(13) starting from eq.(5). Since v =        , eq.(5) is
                                                         dt
equivalent to
                      dx       F
                         = v0 + t                                                (14)
                      dt       m
                                                   F 
and after separation of variables dx =  v 0 +       t dt
                                                   m 

                                                                                        31
and integration,
                                 x         t
                                                      F 
                                 ∫ dx = ∫  v 0 + m t dt
                                                     
                                 0         0

we get
                                                             t
                                                  F t2 
                                [x ] x
                                     0   = v0 t +      
                                           
                                                  m2  0
so as
                                                   1F 2
                                 x = v0 t +           t .
                                                   2m




Using (14), we may arrive at eq.(13) by writing the following Mathematica code




Important remark:
We may derive both equations (5) and (13) by solving the system
                                  dx      
                                      =v 
                                  dt      
                                          
                                   dv     
                                 m     = F
                                   dt     
under the initial conditions x (0) = 0 , v(0) = v 0 .

The corresponding Mathematica code is




We may also eliminate the time t between the two previous equations using




                                                                                 32
and then to solve with respect to x :




                                         r
                          r         r dv
(iv) For the acceleration a we have a =    and so using eq.(7), we get
                                        dt
                               r Fr
                               a= i                            (15).
                                 m
Plot.
1. Velocity in terms of time




2. Displacement in terms of time




                                                                         33
Example 2. A particle of mass m moves in a straight line. A constant
resisting force acts on the particle of magnitude F. If v 0 is the initial speed

of the particle, find
1.     the time taken for the particle to come to rest
2.     the distance travelled.


Solution.




                                              r
                                             dv r
(i) We start with the differential equation m = F which can be expressed as
                                             dt
                                         dv r     r
                                     m      i = −F i
                                         dt


                                                                                   34
or
                            dv
                        m      = −F                                           (1)
                            dt
and after separation of variables,
                        mdv = −Fdt                                            (2)
Since for t = 0, v = v 0 and for the total time of motion t , v = 0 , we integrate

eq.(2) using definite integrals:
                            0            t
                        m ∫ dv = − F ∫ dt
                            v0           t0


                        ⇔ m[v]v= v 0 = −F[t ]0
                                   v=0         t



                        ⇔ m(0 − v 0 ) = − F( t − 0)

                        ⇔ mv 0 = Ft

                                 mv 0
                        ⇔t=                                                   (3)
                                  F




Using Mathematica, we may arrive at (3) following the next steps:
Step1 Solve differential equation (1) taking into account that
v ( 0) = v 0




                                     Ft
Step2 Define function v( t ) = −        + v0
                                     m




Step3 Solve equation v( t ) = 0 with respect to t




                                                                                     35
(ii) We get from (1)
                    dv dx                     dv
                m         = −F    ⇔       m      v = −F                       (4)
                    dx dt                     dx
and after separation of variables,
                                 m vdv = − Fdx                                (5)
The previous equation can be integrated using definite integrals:
                                     0               x
                                 m ∫ vdv= − F∫ dx ,
                                   v0                0

                                          v =0
                                  v2 
                                            = − F[x ]0 ,
                                                     x
                                 m 
                                   2  v=v      0



                                      v2        
                                 m 0 − 0
                                                 = −F(x − 0)
                                                 
                                       2        
from which we get
                                 1
                                   m v0 = F x
                                      2
                                 2
and then
                                         1m 2
                                 x=         v0                         (6)
                                         2F




Using eq.(4), we may arrive at eq.(6) by writing the following Mathematica code




and then determining the arbitrary constant C[1] from the initial condition x ( v 0 ) = 0

In a more detailed calculation, we may use the following steps:
Step1.




Step2.




                                                                                        36
Step3.




We may derive the equation of the orbit and the equation for the speed by solving the
system
                                          dx       
                                              =v 
                                          dt       
                                                   
                                          dv       
                                        m     = − F
                                           dt      
under the initial conditions x (0) = 0 , v(0) = v 0 .




The corresponding Mathematica code is




In case of Maple, enter the system




and then the initial conditions



                                                                                    37
Enter the following and get the answer




The letters in red is the code while the letters in light blue is the answer you get.




In Macsyma, enter the two equations as follows




and then the initial conditions




After that, you enter the code for the solution of the system of the two differential
equations under the sided initial conditions and you get the answer




                                                                                        38
In Scientific WorkPlace, from the menu item Insert , choose Display and then enter
each one of the equations (by pressing Enter after each one equation), as follows:




From the menu item Maple, choose Solve ODE, and then from the appearing
submenu, choose Exact.
The response you get is




Plot
We define the function v( t ) = v 0 − at




We plot the function v( t ) for v 0 = 10 and a = 3,4,5,6,7




                                                                                     39
                                        1
We define the function x ( t ) = v 0 t − at 2
                                        2




We plot the function for v 0 = 10 and a = 2,3,4,5,6




                                                      40
We may also tabulate the functions




                                     41
                                           r
1.1.2. Second Case. The Force F depends on the instantaneous velocity:
                                               r          r
                                               F = F( v ) i .

           For example: F( v) = − av 2 , F( v) = a + bv 2 , F( v) = F − av 2
           where v is the instantaneous speed.


Example 1.
           A particle of mass m moves along a horizontal line such that at t = 0 it has
speed v 0 and it is located at x = 0 . The particle is acted upon by a force which

opposes the motion and has magnitude proportional to the square of the instantaneous
speed : F = − k v 2 , k is the proportionality factor ( k > 0 ).
           Find the speed, the distance and the acceleration of the particle in terms of
time t .
           Express the speed and the acceleration in terms of the distance travelled.


Solution.




                     r
                    dv r
Newton’s equation m    = F can be written as
                    dt
                                       dv r          r
                                   m      i = − kv 2 i
                                       dt
or
                                       dv
                                   m      = −kv 2                         (1)
                                       dt




                                                                                           42
Speed in terms of time.
After separation of variables, we get from (1)
                                   dv
                               m      = − k dt                            (2)
                                   v2
Integrating eq.(2) we get
                                   v              t
                                        dv
                               m∫          = − k ∫ dt ,
                                   v0   v2       0

                                            v
                                  1
                               m −  = −k[ t ]0 ,
                                               t

                                  v  v0

                                 1 1 
                               m − +  = −kt
                                 v v 
                                     0 


                                   1 1    k
                               −    +   =− t                                    (3)
                                   v v0   m

                                1   1  k
                                  =   + t
                                v v0 m

                                1 m + kv 0 t
                                  =
                                v   v0m
                                           m
                                v=                 ⋅ v0 .                       (4)
                                        m + kv 0 t




Using Mathematica, we may arrive at (4) by finding the general solution of the
differential equation (2),




and then determining the arbitrary constant C[1] , by t = 0 , v = v 0 .




                                                                                      43
Distance in terms of time.
            dx
Since v =      , we get from (4)
            dt
                                dx   mv 0
                                   =                                     (5)
                                dt m + kv 0 t

                                  mv 0
and separation variables dx =              dt and integration
                                m + kv 0 t
                                x          t
                                               mv 0
                                ∫ dx = ∫
                                0          0
                                             m + kv 0 t
                                                        dt

we get

                                [ x ]0 =
                                     x     m
                                             [ln(m + kv 0 t )]0t ,
                                           k

                                x=
                                     m
                                       {ln(m + kv 0 t ) − ln m} ,
                                     k
                                     m m + kv 0 t
                                x=     ln                    ,
                                     k    m
and finally
                                     m  kv 0 
                                x=    ln1 +   t                        (6)
                                     k      m 




Using Mathematica, we may arrive at (6) by finding the general solution of the
differential equation (5),




and then determining the arbitrary constant C[1] , by x (0) = 0 .




                                                                                 44
Acceleration in terms of time.
            dv
Since a =      , we get from (4)
            dt


                  d      m           
                                                         −
                                                              d
                                                                 (m + kv 0 t )
                a=                                           dt
                      m + kv t ⋅ v 0  = mv 0
                  dt                                       (m + kv 0 t )2
                                                                               =
                             0        


                              − kv 0
                  = mv 0
                           (m + kv 0 t )2
and finally
                                                                 2
                                                             mkv 0
                                               a=−                                 (7)
                                                        (m + kv 0 t )2


              Velocity in terms of displacement
In order to express the velocity in terms of the displacement, we write (1) as


                                       dv dx
                                  m          = − kv 2
                                       dx dt
                                       dv
                                  m       v = − kv 2
                                       dx
                                       dv
                                  m       = −kv .                                  (8)
                                       dx
After separation of variables and integration, we get from equation (8)
                                       v            x
                                            dv
                                  m∫           = − k ∫ dx
                                       v0
                                             v       0


                                  m[ln v ]v0 = − k[ x ]0
                                              v        x



                                  m(ln v − ln v 0 ) = − kx

                                            v
                                  m ln         = − kx
                                            v0

                                       v    k
                                  ln      =− x
                                       v0   m




                                                                                         45
                                                k
                                     v    − x
                                        =e m
                                     v0

and finally
                                                        k
                                                    −     x
                                     v = v0 ⋅ e         m
                                                              .                  (9).




Using Mathematica, we may arrive at (9) by finding the general solution of the
differential equation (8),




and then determining the arbitrary constant C[1] , by v(0) = v 0 .




          Acceleration in terms of displacement
Using equation (9), we get for the acceleration


                            k
              dv         − x d   k 
        a=       = v0 ⋅ e m     − x =
              dt             dt  m 


                                     k  dx    k
                 = v 0 ⋅ e − kx / m  −     = − v 0 ⋅ e − kx / m v =
                                     m  dt    m



                 =−
                      k
                      m
                                       (                      )
                                                             k 2
                        v 0 ⋅ e − kx / m v 0 ⋅ e − kx / m = − v 0 e − 2 kx / m
                                                             m
and finally
                                             k 2 − 2 kx / m
                                     a=−       v0e                               (10)
                                             m

                                                    Plot
We define the velocity function using eq.(4),




                                                                                        46
            k
where p =     .
            m
We substitute v 0 = 10 and p = 0.1, 0.3, 0.5, 0.7, 0.9




We may also tabulate the values of the velocity function




We now define the function x ( t ) using eq.(6)




                                                           47
and then we plot the function for v 0 = 10 and p = 0.1, 0.3, 0.5, 0.7, 0.9




Finally, we tabulate the values of the function x ( t )




Question 1. Express the time needed by the particle to develop a given velocity v.
Answer.   We get from (3)
                                 k   1 1
                                   t= −
                                 m   v v0

which is equivalent to

                                      m   1 1     
                                 t=        −
                                          v v     
                                                   
                                      k       0   
Question 2. Express the time needed by the particle to cover a given distance x.
Answer. We get from (6)

                                 k        kv 
                                   x = ln1 + 0 t 
                                 m          m 


                                                                                     48
and after inversion,
                                  k
                                      x     kv 0
                                em = 1+          t⇔
                                             m
                                                 k
                                      kv 0         x
                                ⇔          t = e m −1 ⇔
                                       m

                                      m       mx
                                                k
                                                    
                                ⇔t =          e − 1
                                     kv 0          
                                                   



Example 2. An object of mass m moves along a straight line under a force of
                       r       r
constant magnitude F : F = F ⋅ i . We assume that there is a resisting force acting on
the object which is proportional to the square of the instantaneous speed :
 r           r
 FR = − kv 2 i where k is the proportionality factor. Show that the distance traveled in

                                                                 m F − kv1 2
going from a point of speed v1 to a point of speed v 2 is s =      ln        .
                                                                 2k F − kv 2
                                                                           2




                                                      r               r
Solution. The total force acting on the particle is F = ( F − k v 2 ) i .
                     r
                    dv r
Newton’s equation m    = F can be written as
                    dt
                            dv r                 r
                          m     i = (F − k v 2 ) i
                             dt
or




                                                                                         49
                                         dv
                                    m       = F − kv 2
                                         dt
                                          dv dx
                                  ⇔ m           = F − kv 2
                                          dx dt
                                          dv
                                  ⇔ m        v = F − kv 2                    (1)
                                          dx
Separating the variables and integration we get from the previous equation
                             v2                  x2
                                     v
                        m∫                   dv = ∫ dx
                             v1   F − kv 2       x1


                         ⇔ −
                                   m
                                   2k
                                         [
                                      ln(F − kv 2 )      ]
                                                         v2
                                                         v1
                                                                    x
                                                              = [x] x 2 ⇔
                                                                   1


                         ⇔ −
                                   m
                                   2k
                                         [                             ]
                                      ln(F − kv 2 ) − ln(F − kv1 ) = x 2 − x1 ⇔
                                                2
                                                               2



                              m F − kv1 2
                        ⇔       ln        =s                                       (2)
                              2k F − kv 2
                                        2




Using Mathematica, we may arrive at (2) by finding the general solution of the
differential equation (1),




and then determining the arbitrary constant C[1] , by x ( v 0 ) = 0 .

The code appears in the following




                                                                                    50
Example 3. A small car moves along a horizontal track. There is a resisting
                                                 r              r
force acting on the car which is given by FR = − (λ + µ v 2 ) i where λ > 0, µ > 0 .

If the initial speed is v 0 calculate

(i)     The time the car needs to come to rest.




                               r
                              dv r
Solution. Newton’s equation m    = F , can be written as
                              dt
                              dv r                  r
                            m    i = − (λ + µ v 2 ) i
                              dt
or
                                     dv
                                 m      = − (λ + µ v 2 )                     (1)
                                     dt
(i)     Separating the variables and integration, we get from equation (1)
                                     0                   t
                                            dv
                                 m∫                  = − ∫ dt
                                     v0   λ + µv 2       0

                                             0
                                          m      dv
                                  ⇔
                                          µv∫ λ 2 = −t                             (2)
                                             0   +v
                                               µ
Under the change of variables

                               λ                     λ
                         v=      ⋅y,        dv =       dy ,
                               µ                     µ

                                              µ
                         v = v0 → y =           v0 , v = 0 → y = 0 .
                                              λ



                                                                                       51
equation (2) becomes

                                                    λ
                                                      dy
                                m
                                       0
                                                    µ
                                       ∫                       = −t
                                µ               λ
                                     µ / λ v0     (1 + y 2 )
                                                µ

                                       m µ
                                 ⇔         [arctan y]0 µ / λ v = − t
                                       µ λ                    0


                                  m                      µ 
                                       arctan 0 − arctan       
                                                          λ v 0  = − t
                                  µλ   
                                                               
                                       m         µ 
                             ⇔ t=         arctan      
                                                 λ v0                     (3)
                                       µλ             



Using Mathematica, we may arrive at (3) by finding the general solution of the
differential equation (1),




and then determining the arbitrary constant C[1] , by t ( v 0 ) = 0 .

In the following, we arrive at (3), using a different procedure
Step1.




Step2.




                                                                                  52
Step3.




Using Reduce,




(ii)     Equation (1) can be transformed into
                                   dv dx
                               m         = − (λ + µ ν 2 )
                                   dx dt
                                          dv
                                ⇔ m          ⋅ v = − (λ + µ ν 2 )   (4)
                                          dx
After separation of variables and integration, we get
                                   0                     x
                                           v
                               m∫                  dv = − ∫ dx
                                   v0   λ + µν 2          0




                                                                          53
                                 ⇔
                                     m
                                     2µ
                                         [
                                        ln(λ + µν 2 )   ]0
                                                         v0   = − [ x ]0
                                                                       x


                                     m
                                 ⇔      { ln λ − ln(λ + µν 0 )} = − x
                                                           2
                                     2µ

                                   m       λ + µν0
                                                 2
                                 ⇔    ⋅ ln         =x
                                   2µ         λ
                                        m  µ 2
                                ⇔x=       ln1 + ν 0                            (5)
                                        2µ  λ 




Using Mathematica, we may arrive at (5) by finding the general solution of the
differential equation (4),




and then determining the arbitrary constant C[1] , by x ( v 0 ) = 0 .

We may arrive at (5) using the following steps:
Step1.




Step2.




Step3.




                                                                                  54
Example 4. A small object of mass m moves along a horizontal axis. The initial
velocity of the object is v 0 . A resisting force acting on the particle is proportional to
                                              r             r
the cube of the instantaneous speed : FR = − k v 3 i where k (k > 0) is the

                                                                      1
proportionality factor. We suppose that after time τ the speed is       v0 .
                                                                      2
                                      1
(i)     Prove that the speed is         v0 after time 5 τ .
                                      4
                                                                                 1
(ii)    Calculate the distance traveled by the object in reaching the speeds       v0 and
                                                                                 2
        1
          v0 .
        4




Solution.
                                  r
                                 dv r                       dv r          r
Newton’s differential equation m    = F can be written as m    i = − k v3 i
                                 dt                         dt
or
                                          dv
                                     m       = − k v3                                   (1)
                                          dt
                                                                     1
(i)     We shall first determine the time τ at which the speed is      v 0 . We separate the
                                                                     2
        variables in equation (1) and integrate :
                            v0 / 2                τ
                                     dv
                        m     ∫           = − k ∫ dt ⇔
                             v0      v3           0

                                           v /2
                           1  0                           τ
                        m − 2                   = − k[ t ]0 ⇔
                           2v  v 0




                                                                                          55
                              1      1 
                        m −        + 2  = − kτ ⇔
                          2(v / 2)2 2 v 
                             0         0

                          2     1 
                        m − 2 + 2  = − kτ ⇔
                          v       
                          0 2v 0 
                             3
                        m     2
                                  = kτ         ⇔
                            2v0
                             3m
                        τ=      2
                                                                             (2)
                             2kv0



In order to arrive at (2), we use the following steps, using differential
equation (1) and the initial condition v(0) = v 0

Step1.




Step2.




Step3.




Integrating equation (1) between the limits v 0 and v 0 / 4 (respectively 0 and τ ′ ) we

have
                                      v0 / 4              τ′
                                               dv
                                  m     ∫           = − k ∫ dt ,
                                       v0      v3         0




                                                                                       56
                                                v /4
                                   1  0
                                m − 2                = − k [ t ]0′ ,
                                                                  τ
                                   2v  v0

                                         1         1 
                                m −              + 2  = − k τ′
                                   2 (v 0 / 4 )
                                                2
                                                  v0 
                                                      
                                   8     1 
                                m  − 2 + 2  = − k τ′ ,
                                   v       
                                   0 2v 0 
                                    15
                                m      2
                                         = kτ ′ ,
                                    2v 0
and finally
                                         3m
                                τ′ = 5      2
                                                                               (3)
                                         2kv0
A comparison of (2) and (3) gives us τ′ = 5τ                                           (4).




In order to arrive at (3), we use the following steps, using differential
equation (1) and the initial condition v(0) = v 0

Step1.




Step2.




Step3.




(ii)     In order to find the distances travelled by the particle in reaching the speeds



                                                                                           57
        1        1
          v 0 and v 0 , we start using the differential equation (1) which can be
        2        4
       transformed as




                   dv dx            dv
               m         = −kv 3 ⇔ m v = − kv 3
                   dx dt            dx
                                             dv
                                ⇔ m             = − kv 2 .                  (5)
                                             dx


Separating the variables and integration, we get from equation (5)
                                   dv
                               m      = −kdx ⇔
                                   v2
                                   v0 / 2               x
                                            dv
                               m    ∫
                                    v0      v2
                                               = −k ∫ dx ⇔
                                                    0

                                             v0 / 2
                                  1
                               m −                  = − k[ x ]0
                                                                x

                                  v  v0

                                    1    1 
                                 v / 2 + v  = − kx ⇔
                               m −          
                                   0      0 


                                 1 
                               m −  = − kx ⇔
                                 v 
                                   0 


                                          m
                               x=             ⇔                             (6a)
                                         kv 0

                                         2  3m 
                               x=         v0       ⇔
                                         3  2kv 0 
                                             
                                                 2
                                                   
                                         2
                               x=          v0 τ .                           (6)
                                         3


Similarly integrating equation (5) between the limits v 0 and v 0 / 4 we find


                                   v0 / 4               x′
                                            dv
                               m    ∫
                                    v0      v2
                                               = −k ∫ dx ⇔
                                                    0




                                                                                    58
                                          v /4
                                  1
                                        0

                               m −             = − k[ x ]0 ′ ⇔
                                                           x

                                  v  v0

                                    1    1 
                                 v / 4 + v  = − kx ′
                               m −          
                                   0      0 




                                3m
                                   = kx ′ ⇔
                                v0

                                       3m
                                x′ =        ⇔                                 (7a)
                                       kv 0

                                            3m      
                                x ′ = 2v 0 
                                            2kv 2    ⇔
                                                     
                                                0   
                                x ′ = 2v 0 τ .                                (7)




In order to arrive at (6a) and (7a), we use the following steps, using differential
equation (5) and the initial condition v(0) = v 0

Step1.




Step2.




Step3.




Step4.




                                                                                      59
                                        Example 5.
A small object moves in a straight path such that the time taken to reach a distance x from the
origin is given by
                                   t = Ax 2 + Bx + C                                 (1)
where A,B,C are constants (A > 0) .
Prove that the force acting on the particle is proportional to the cube of the instantaneous
speed:

                                   F = − k v3        ( k > 0)                        (2)

Solution. The total force acting on the particle is given by
                                             d2x
                                   F=m                                               (3)
                                             dt 2
Therefore, we have to find the second derivative of the displacement with respect to time,
using the initial hypothesis (1). We first differentiate eq.(1) with respect to t:
                  d         d
                     ( t ) = (Ax 2 + Bx + C)
                  dt        dt
                          d                   dx
                 ⇔1=        (Ax 2 + Bx + C) ⋅
                         dx                   dt
                                        dx
                 ⇔ 1 = (2Ax + B)                                                     (4)
                                        dt
We differentiate further eq.(4) with respect to t:
                  d       d         dx 
                     (1) = (2Ax + B) 
                  dt      dt        dt 

                       d          dx            d  dx 
                 ⇔ 0 =  (2Ax + B)    + (2Ax + B)  
                        dt        dt            dt  dt 

                       d         dx  dx            d2x
                 ⇔ 0 =  (2Ax + B)       + (2Ax + B) 2
                        dx       dt  dt            dt
                                    2
                          dx 
                                                      2
                                        d x
                 ⇔ 0 = 2A  + (2Ax + B) 2
                          dt          dt



                                                                                               60
                                                   d2x
                   ⇔ 0 = 2Av 2 + (2Ax + B)                             (5)
                                                   dt 2
The factor 2Ax + B appearing in (5) can be expressed in terms of the instantaneous velocity
v, using eq.(4):
                                                   1
                                       2Ax + B =                               (6)
                                                   v
We also have from Eq.(3),

                                       d2x F
                                           =                                   (7)
                                       dt 2 m
Using (6) and (7), we obtain from (5):
                                            1 F
                         0 = 2Av 2 +         ⋅
                                            v m
                         ⇔ 0 = 2Amv 3 + F
                         ⇔ F = −2Amv 3

                         ⇔ F = − k v3 ,
where k = 2AM > 0 .
                          dv
2nd Method. Since m          = F , we have
                          dt
                             dv
                         m      = dt
                             F
                             v          t
                             dv
                         m∫     = ∫ dt = Ax 2 + Bx + C                         (8)
                          v0
                             F 0

and differentiating with respect to v,
                                   v
                           d dv d
                              ∫ = (Ax + Bx + C)
                                      2
                         m
                           dv v0 F dv

                                   d   d                 dx
                         ⇔m          =   (Ax 2 + Bx + C)
                                   F dx                  dv
                                 m             dx
                         ⇔         = (2Ax + B)                                 (9)
                                 F             dv
On the other hand,
                             dv      dv dx
                         m      =F⇔m       =F
                             dt      dx dt
                                   dv
                         ⇔m           v=F
                                   dx



                                                                                         61
                                dx mv
                            ⇔      =                                                 (10)
                                dv   F
Using (9) and (10) we find
                            m             mv
                              = (2Ax + B)
                            F             F


                                          1
                            ⇔ 2Ax + B =                                      (11)
                                          v
                                  1    1
                            ⇔ x =  − B                                     (12)
                                  v    2A
Substituting x, as given by Eq.(12), in Eq.(8), we get
                                                2
                     v
                     dv      1    1      1       1
                 m∫     = A  − B      + B v − B  2A + C
                  v0
                     F       v    2A            
We differentiate with respect to v the previous equation and we get
                     1     1      1     1  B  1 
                 m     =A    2
                               ⋅ 2 − B  − 2  + − 2 
                     F    4A      v     v  2A  v 
                         m   1 1     1  B  1 
                 ⇔         =    − B  − 2  + − 2 
                         F 2A  v     v  2A  v 
                         m     1 1
                 ⇔         =−
                         F    2A v 3
                 ⇔ F = −2Amv 3
                 ⇔ F = − kv 3 ,
where
                 k = 2Am




                                       Example 6.
A small object moves in a straight path such that the force acting on the particle
is given by
                                  r         r
                                  F = −kv 3 i       ( k > 0)
where v is the instantaneous speed. Prove that the time t taken to reach a
distance x is given by
                                  t = Ax 2 + Bx + C
where A,B,C, are constants (A > 0) .


                                                                                            62
Solution: Since
                                                 r
                                                dv r
                                            m      =F
                                                dt
we get
                                                dv
                                            m      = − kv 3
                                                dt
                                                dv dx
                                            m         = −kv 3
                                                dx dt
                                                dv
                                            m      v = − kv 3
                                                dx
                                                dv
                                            m      = − kv 2         (1)
                                                dx
We integrate eq.(1) and we get
                                    v             x
                                     dv
                                 m ∫ 2 = −k ∫ dx
                                  v0 v     x0

                                            v
                                    1
                                 m −  = −k ( x − x 0 )
                                    v  v0

                                   1 1 
                                 m − +  = − k ( x − x 0 )
                                   v v 
                                       0 


                                  1   1  k
                                    =   + (x − x 0 )
                                  v v0 m

                                  1 k(x − x 0 ) + m
                                    =
                                  v     mv 0

                                                mv 0
                                 v=                                 (2)
                                        kv 0 ( x − x 0 ) + m
            dx
Since v =      , eq.(2) is equivalent to
            dt
                         dx         mv 0
                            =
                         dt kv 0 ( x − x 0 ) + m
which can be solved by separation of variables
                         [kv 0 ( x − x 0 ) + m]dx = mv 0 dt
and integration
                         x                                      t

                         ∫ [kv 0 (x − x 0 ) + m]dx = mv 0 ∫ dt
                         x0                                    0




                                                                          63
                             x                         x
                        kv 0 ∫ xdx − (kv 0 x 0 − m) ∫ dx = mv 0 t
                             x0                       x0

                                    x
                             x2 
                        kv 0   (kv 0 x 0 − m)( x − x 0 ) = mv 0 t
                              2  x0
                        1
                          kv 0 ( x 2 − x 0 ) − (kv 0 x 0 − m) x + (kv 0 x 0 − m) x 0 = mv 0 t
                                         2

                        2
                        1           1
                          kv 0 x 2 − kv 0 x 0 − (kv 0 x 0 − m) x + (kv 0 x 0 − m) x 0 = mv 0 t
                                            2

                        2           2
                        1                              1
                          kv 0 x 2 − (kv 0 x 0 − m) x + kv 0 x 0 − mx 0 = mv 0 t
                                                               2

                        2                              2
                              k 2 m − kv 0 x 0   kv x 2 − 2mx 0
                        t=      x +            x+ 0 0
                             2m     mv 0             mv 0

                        t = Ax 2 + Bx + C


where
                               k
                        A=
                              2m
                             m − kv 0 x 0
                        B=
                               mv 0

                             kv 0 x 0 − 2mx 0
                                    2
                        C=
                                   mv 0




                                         r                      r         r
1.1.3. Third Case. The total force F         depends on time : F = F( t ) i


Consider a particle of mass m moving on a straight line under the action of a
force F which depends only on the time t at which the force is measured.
The equation of motion becomes
                                  m&& = F( t ) .
                                   x

            dx         d2t
where x =
      &        and && = 2 is Newton’s notation for the time derivatives.
                   x
            dt         dt
To integrate this equation you will find it convenient to write it in the form




                                                                                                 64
                                      &
                                     dx F( t )
                                        =      .
                                     dt   m


Integrating with respect to time gives


                                  F( t )
                           x=∫
                           &             dt    or    x = f ( t ) + c1
                                                     &
                                   m


where f ( t ) is a particular value of the integral and c1 is a constant of integration.

Rewriting this equation in the form


                                     dx
                                        = f ( t ) + c1
                                     dt


a second integration with respect to time gives


                 x = ∫ f ( t )dt + c1 t       or         x = g ( t ) + c1 t + c 2


where g ( t ) is a particular value of the integral        ∫ f (t )dt   and c 2 is a second constant of

integration. With experience it becomes unnecessary to rewrite derivatives explicitly in
           d
the form      before integrating with respect to time.
           dt



        The displacement of the particle at time t has been found very easily in this case.
The function x ( t ) includes two arbitrary constants and describes all possible motions
of the particle. To find the actual motion of the particle we require some initial
conditions.
        Suppose that at time t = 0 the particle is observed to pass the point x 0 with

velocity v 0 . Here “the point x 0 ” means the point with displacement x 0 relative to the

                                                                                    &
given origin O. Substituting this information into the equations obtained above for x
and x gives
                           v 0 = f (0) + c1 and x 0 = g (0) + c 2




                                                                                                      65
These two equations determine the constants c1 and c 2 uniquely. With hindsight it

would have been rather clever to choose the particular values f ( t ) and g ( t ) of the

integrals to satisfy f (0) = 0 and g (0) = 0 , the constants of integration could then have
been interpreted directly as the initial velocity and initial displacement. This can be
done by evaluating the two definite integrals


                         t                                      t
                           F( t )
                 f (t) = ∫        dt         and      g( t ) = ∫ f ( t )dt .
                         0
                            m                                   0




        It is very important to understand that there is no unique choice for the initial
conditions. This can be seen quite easily by considering whether the above constants of
integration can be determined without knowing both the initial displacement and
velocity of the particle explicitly. Suppose that at time t = 0 the particle is observed to
pass the point x 1 . Substituting this information into the equation for x gives


                 x 0 = g ( 0) + c 2    and     x 1 = g (t 1 ) + c 1 t 1 + c 2


        These two equations can be solved algebraically to give c1 and c 2 and hence to

determine the actual motion of the particle uniquely.




The common property of all the different possible initial conditions is that from each
the initial displacement and velocity can be determined either directly or by using one
or both of the integrals of the equation of motion.



Example 1.
A particle of mass m moves on a straight line under the action of a force
F( t ) = F0 ⋅ sin(pt ) , where F0 and p are positive constants. The velocity of the

particle at time t = 0 , relative to a fixed origin O, is v 0 . Show that the particle

                                                       F0
will move towards x = +∞ if and only if v 0 >             .
                                                       mp




                                                                                            66
                                                       F0
Describe the motion of the particle when v 0 = −          .
                                                       mp
Solution.
The equation of motion
                                           m&& = F0 sin pt
                                            x
can be integrated twice to give
                                                    F0
                                           mx = −
                                            &          cos pt + c1
                                                    p
and
                                                    F0
                                           mx = −      sin pt + c1 t + c 2 .
                                                    p2
             F0
The term −      sin pt in the expression for the displacement of the particle is bounded,
             p2
in fact it represents an oscillation. As t increases this will become negligible compared
to c1 t and so the particle will move towards x = +∞ if and only if c1 > 0 .

Now x = v 0 when t = 0 so that
    &

                                             F0
                                  mv 0 = −      + c1
                                             p
and the inequality c1 > 0 can be written as

                                           F0
                                  mv 0 +      >0
                                           p
                     F0
from which v 0 > −      .
                     mp
Similarly the particle will move towards x = −∞ if and only if c1 < 0 which leads to

                                           F0
                                  v0 < −      .
                                           mp
               F0
When v 0 = −      the constant of integration c1 becomes zero so that
               mp
                                           F0
                                  mx = −      sin pt + c 2
                                           p2
and the motion is purely oscillatory.



Exercises


                                                                                            67
1. The particle of Example 1 is observed to pass the points x 0 and x 1 at times t = 0

              2π
    and t =      . Find alternative conditions for the different motions discussed in the
               p
    example in terms of x 1 and x 2 .
2. A particle of mass m is moving on a straight line under the action of an
    exponentially decreasing force F = F0 e − λt , where F0 and λ are positive constants.

    The particle passes the point x 0 with velocity v 0 at time t = 0 . Obtain an

    expression for the displacement of the particle at time t and sketch a graph of the
                                                                         F0
    displacement x against time for the special case when v 0 = −           .
                                                                         mλ
              − at
3. F = kte




1.1.4. Fourth Case.         Force depends on the replacement : F = F( x ) .


In this case Newton’s equation can be written as


                                        d2x
                                    m        = F( x )              (1)
                                        dt 2


We consider the initial conditions : t = 0 , x = x 0 and v = v 0 .

                                                    dx
Multiply both sides of the equation (1) by 2           .
                                                    dt
We find


                                                              2
                       dx d 2 x           dx     d  dx             dx
                       dt dt 2  = 2F( x ) dt ⇔ m dt  dt  = 2F( x ) dt
                     m 2        
                                                     


and then integrating from 0 to t we find


                       t        2         x
                         d  dx 
                     m ∫   dt = 2 ∫ F( x )dx
                       0
                         dt  dt  x0




                                                                                            68
                 1  dx         dx  
                           2            2      x
                                            
                ⇔ m   −    = ∫ F( x )dx
                 2  dt  t = t  dt  t =0  x 0
                                            


                    1  dx       x
                              2
                               2
                ⇔    m   − v 0  = ∫ F( x )dx
                    2  dt 
                                  x0
                                  


                    1  dx               1
                            2   x
                ⇔    m  + − ∫ F( x )dx  = mv 0 = const.
                                                 2

                    2  dt   x 0
                                          2
                                          




The first term of this equation is the kinetic energy, the second term the potential
energy of the moving point. From this equation it follows that the sum of the
kinetic and potential energy remains constant through the time of motion.
Put :
                                            x
                                 U( x ) = − ∫ F( x )dx
                                           x0


from which there follows
                                 dU( x )
                                         = − F( x )
                                  dx
where U ( x ) = potential function or potential.




1.2. Vertical Motion.

In this case we consider two subcases :
                                                      r
a) Vertical upward (necessarily with initial velocity v 0 ≠ 0 ).




                                                                                       69
                                                 r          r
b) Vertical downward (where the initial velocity v 0 = 0 or v 0 ≠ 0 ).




1.2.1. Upward Vertical Motion.


Example 1.



                                                                         70
A small object of mass m is thrown vertically upward with speed v 0 . Assume that air

resistance is proportional to the instantaneous speed. Find
(i)    the maximum height reached by the object.
(ii)   the time taken to reach the maximum height.


Solution.




                                  r
                                 dv r
Newton’s differential equation m    = F can be written as
                                 dt
                          r
                         dv r r
                       m    = w+R.                                               (1)
                         dt
                r                                r        r
where w = − mgk is the weight of the object and R = −avk is air’s
resistance (a is the proportionality factor and v is the instantaneous speed).
                                      dv r              r
Equation (1) can so be written as m      k = −(mg + av )k
                                      dt
or

                              = −(mg + av )
                           dv
                       m                                                         (2).
                           dt
(i)    Equation (2) can be transformed as

                                 = −(mg + av ) ,
                           dv dx
                       m
                           dx dt



                              v = −(mg + av ) .
                           dv
                       m                                                         (3)
                           dx
Separation of variables and integration, gives us



                                                                                        71
                             0                 x
                                  v
                        m∫              dv = − ∫ dx ,
                             v0
                                mg + av        0

                              0
                        m     av
                          ∫ mg + avdv = −[ x]0
                                             x

                        a v0

                        m        mg 
                          0

                          ∫ 1 − mg + av dv = −x ,
                             
                        a v0            
                                         
                              0           0
                        m       m 2g      a
                          ∫ dv − a 2 v∫ mg + av dv = − x
                        a v0           0



                        m 0 m 2g
                          [ v] v0 − 2 [ln(mg + av)]v0 = − x
                                                   0

                        a           a
                                         2
                        m
                          (0 − v 0 ) − m 2g {ln(mg) − ln(mg + av 0 )} = − x ,
                        a              a
                        mv 0 m 2 g mg + av 0
                            − 2 ln           =x
                         a    a      mg
and finally
                           mv 0 m 2 g  av 0 
                        x=     − 2 ln1 +
                                            .                                 (4)
                            a    a       mg 
                                             




Using Mathematica, we may arrive at (4) by finding the general solution of the
differential equation (3),




and then determining the arbitrary constant C[1] , by x ( v 0 ) = 0 .

The steps appear in the following:




Step1.



                                                                                      72
Step2.




Step3.




(ii)     From equation (2) we get
                             0              t
                        m    adv
                          ∫ mg + av = −∫ dt ,
                        a v0           0


                        m
                          [ln(mg + av]0 0 = −[ t ]0t
                                      v
                        a
                        m
                          {ln(mg) − ln(mg + av 0 )} = − t
                        a
                             m mg + av 0
                        t=     ln
                             a    mg

                             m  av 0 
                        t=    ln1 +                                   (5)
                             a     mg 
                                        




Using Mathematica, we may arrive at (5) by finding the general solution of the
differential equation (2),




and then determining the arbitrary constant C[1] , by t ( v 0 ) = 0 .

The steps are the following


Step1.


                                                                                 73
Step2.




Step3.




Example 2.




A small stone of mass m is thrown vertically upward from the earth’s surface with
initial speed v 0 . Calculate

(i)      The position of the stone in terms of the time.
(ii)     The maximum height reached.
(iii)    The time taken to reach the highest point.


                                                                                    74
(iv)    The speed in terms of its distance z from the origin O.

                                                                     r      r
Solution. The total force acting on the stone is the gravity force : F = −mgk .
                        r
                       dv r
Newton’s equation m       = F becomes
                       dt
                                 dv r        r
                               m k = − mgk or
                                 dt
                                 dv
                                    = −g                                    (1).
                                 dt
(i)     Separating variables and integration, we get from (1) :
                                 v            t

                                 ∫ dv = −g ∫ dt ,
                                 v0           0


                                 [ v] v0 = −g[ t ]0 ,
                                      v
                                                  t




                                 v − v 0 = −gt

                                 v = v 0 − gt .                             (2)


            dz                         dz
Since v =      , equation (2) gives us    = v 0 − gt and after separation
            dt                         dt
of variables and integration,
                                 z        t

                                 ∫ dz = ∫ (v 0 − gt )dt ,
                                 0        0

                                                        t
                                                 t2 
                                 [z] =  v 0 t − g  ,
                                      z
                                      0
                                                 2 0

                                            1
                                 z = v 0 t − gt 2 .                         (3)
                                            2
We may get eqs.(2) and (3) solving the system
                                  dz      
                                     =v
                                  dt      
                                          
                                 dv       
                                     = −g 
                                 dt       
under the initial conditions z(0) = 0 , v(0) = v 0 .



                                                                                   75
The above system can be solved using Mathematica. The code appears in the
following




(ii) From equation (1) we get
                                dv dz
                                      = −g        or
                                dz dt
                                dv
                                   v = −g                                (4)
                                dz




and after separation of variables and integration
                                0             h

                                ∫ vdv = −g ∫ dz
                                v0            0


(where it is obvious that at the maximum height h we have v = 0 )
                                      0
                                v2 
                                  = −g[z]0
                                           z

                                 2 v    0


                                   2
                                  v0
                                −    = −gh
                                  2
and so
                                      2
                                     v0
                                h=      .
                                     2g




Using Mathematica, we may arrive at (5) by finding the general solution of the
differential equation (4),




                                                                                 76
and then determining the arbitrary constant C[1] , by z( v 0 ) = 0 .

We may incorporate the initial condition in the code. In this case we get the result




We then define the function z( v)




and h is the value z(0) of the function




                                     0               t
   (iii)      From eq.(1) we get     ∫ dv = −g ∫ dt where it is obvious that at the highest
                                     v0              0


              point the velocity is v = 0.
We get

                                  [v]0v   0
                                              = −g[ t ]0 ,
                                                       t



                                   0 − v 0 = − g ( t − 0) ,

                                   v 0 = gt

and finally
                                                     v0
                                                t=      .
                                                     g




Using Mathematica, we may arrive at (6) by finding the solution of the
differential equation (1),




                                                                                              77
    (iv)      From eq.(4) we get after separation of variables and integration
                                 v            z

                                 ∫ v dv = −g ∫ dz ,
                                 v0           0

                                      v
                                 v2 
                                   = −g[z ]0 ,
                                             z

                                  2  v0
                                          2
                                 v2 v0
                                   −   = −gz ,
                                 2   2
                                 v 2 − v 0 = −2gz ,
                                          2




and finally

                                 v 2 = v 0 − 2gz
                                          2
                                                                                 (7)




Using Mathematica, we may arrive at (7) by finding the solution of the
differential equation (4), under the condition z( v 0 ) = 0




1.2.2. Downward Vertical Motion.




                                                                                       78
Example 1. A parachutist having weight of magnitude mg              leaves the plane with initial
            r
velocity v 0 . If the air resistance acting on the parachute is proportional to the instantaneous
        r          r
speed, FR = −a v k ( a > 0 ) and a is the proportionality factor, calculate

(i) The speed (ii) The distance traveled (iii) The acceleration at any time t > 0 , (iv) The
speed as a function of the distance traveled and (v) The limiting speed of the parachutist.




                                                    r               r
Solution. The total force acting on the particle is F = ( mg − av ) k .
                     r
                    dv r
Newton’s equation m    = F becomes
                    dt
                               dv r         r
                             m k = (mg − av)k
                               dt
or
                                      dv
                                  m      = mg − av                                    (1).
                                      dt
(i)     Separation of variables and integration, equation (1) becomes
                                      v              t
                                           dv
                                  m∫
                                         mg − av ∫
                                                = dt ,
                                      v0         0




                                  −
                                      m
                                        [ln(mg − av)]v0 = [ t ]0t
                                                     v
                                      a

                                  −
                                      m
                                        [ln(mg − av) ln(mg − av 0 )] = t ,
                                      a



                                                                                                79
                                         m mg − av
                                    −     ln         =t
                                         a mg − av 0
                                         mg − av
                                    ln             = −at / m ,
                                         mg − av 0
                                    mg − av
                                              = e −at / m
                                    mg − av 0

                                    mg − av = (mg − av 0 )e − at / m ,

                                    av = mg − (mg − av 0 )e − at / m

                                          mg  mg       
                                    v=      −    − v 0 e − at / m                (2)
                                          a  a         




Using Mathematica, we may arrive at (2) by finding the general solution of the differential
                                    dt   m
equation (1), written in the form      =
                                    dv mg − av




and then determining the arbitrary constant C[1] , by t ( v 0 ) = 0 .

We may incorporate the initial condition in the code. In this case we get the result




and then solve with respect to v :




            dz
Since v =      we get from (2)
            dt
                          dz mg  mg       
                             =   −  − v 0 e −at / m                              (3)
                          dt   a  a       



                                                                                          80
and after separation of variables and integration,


                         z          t
                                      mg  mg                 
                         ∫ dz = ∫        −   − v 0 e −at / m dt ,
                         0        0
                                        a  a                  
                                                                          t
                                mgt  mg        m            
                         [z] = 
                             z
                             0      −    − v 0  − e − at / m 
                                a    a         a            0

                         z=
                                 mgt  m  mg
                                    + 
                                                    
                                                              (
                                              − v 0  e −at / m − 1  )
                                  a  a  a         
and finally

                         z=
                                 mg
                                 a
                                       m
                                    t +  v0 −
                                       a
                                               mg 
                                               a 
                                                   1− e (
                                                         − at / m
                                                                  )
                                                                  .             (4)




Using Mathematica, we may arrive at (4) by finding the general solution of the differential
equation (3),




and then determining the arbitrary constant C[1] , by z(0) = 0 .
If we incorporate the initial condition in the code, we get




(ii)    From equation (1) we get
                                            dv dz
                                        m         = mg − av
                                            dz dt




                                                                                          81
                                        dv
                                    m      v = mg − av .                        (5)
                                        dz


After separation of variables and integration, we get from equation (5) :


                                        v                z
                                             v
                                    m∫             dv = ∫ dz .                  (6)
                                        v0
                                           mg − av      0

Since
          v        1 − av
                =−           =
        mg − av    a mg − av

                       1 (mg − av) − mg    1    mg 
                  =−                    = − 1 −
                                             mg − av  =
                                                      
                       a    mg − av        a         
                    1 mg   1       1 mg − a
                  =− +           =− − 2
                    a  a mg − av   a a mg − av
equation (6) becomes

                                              −a
                                v             v
                             m        mg
                         −     ∫ dv − a 2 v∫ mg − av dv = z ,
                             a v0           0




                               [ v] v0 − 2 [ln(mg − av)]v0 = z
                             m v mg
                         −
                                                        v

                             a          a

                         −
                             m
                               (v − v 0 ) − mg ln mg − av = z ,
                             a              a2   mg − av 0

                                              mg − av 0
                         m
                           (v 0 − v ) + mg ln           =z                      (7)
                         a              a 2
                                              mg − av


The previous equation expresses v as a function of z implicitly.




Using Mathematica, we may arrive at (7) by finding the general solution of the differential
equation (5),




                                                                                          82
and then determining the arbitrary constant C[1] , by z( v 0 ) = 0 .

We may also incorporate the initial condition in the code. In this case we find




(v) The limiting speed the parachutist acquires can be found by the condition:
                                           dv
                 acceleration = 0 ⇔           =0
                                           dt
                                                     mg
and because of equation (1), we get v =                 which is the limiting velocity.
                                                      a


Example 2. A parachutist falls from rest and acquires a limiting speed of v lim . Assuming
that air resistance is proportional to the instantaneous speed, determine how long it takes to
reach the speed of v ( v < v lim ).
Solution. We follow the same line of reasoning as in previous example in reaching equation
(2) with v 0 = 0 :
                                                      at
                              mg mg − m
                           v=   −   e                                               (1)
                              a   a
Since, again, by previous example, the limiting velocity is
                                      mg
                           v lim =                                                  (2)
                                      a
we get a system of two equations : (1) and (2). We solve equation (1) with
respect to t: We have that
                                                at
                                 mg − m mg
                           (1) ⇔   e =     −v⇔
                                 a      a


                                    at
                                −               a
                           ⇔e       m
                                         = 1−      v⇔
                                                mg

                                 at         a 
                           ⇔−            mg v  ⇔
                                    = ln1 −   
                                 m            



                                                                                                 83
                                  m         a 
                        ⇔t=−        ⋅ ln1 −
                                         mg v 
                                                                         (3)
                                  a           
From equation (2) we get
                                 m v lim
                                   =                                      (4)
                                 a   g
Using (2) and (4) in equation (3) we get

                                       v lim         v     
                                 t=−         ⋅ ln1 −
                                                  v        .
                                                            
                                        g            lim   



Example 3. A small object of mass m is dropped from a height h above the ground,
with an initial velocity of magnitude v 0 . Prove that air resistance is negligible, then the

                                              v 0 + 2gh − v 0
                                                2

object will reach the ground in time t =                         with speed
                                                    g

v = v 0 + 2gh .
      2




Solution.




The total force acting on the object is the gravity force :
                                 r      r
                                 Fw = mgk .
                     r
                    dv r                            dv r     r
Newton’s equation m    =F         becomes       m      k = mgk
                    dt                              dt
or
                                 dv
                                    =g                             (1)
                                 dt




                                                                                           84
From equation (1) we get

                                   dv dz      dv
                                         =g ⇔    v=g
                                   dz dt      dz

and after separation of variables and integration,

                                                 v
                  v          z      v2 
                 ∫v0 vdv = g ∫ dz ⇔   = g[z]0 ⇔
                                              z
                              0
                                     2  v0



                                    2
                           v2 v0
                         ⇔     −      = gz
                            2     2
                         ⇔ v 2 − v 0 = 2gz ⇔
                                   2


                         ⇔ v 2 = v 0 + 2gz
                                   2


                         ⇔ v = v 0 + 2gz
                                 2
                                                                             (2)


Equation (2) gives the speed at any point between O and the ground.
            dz
Since v =      , we get from (2)
            dt
                                   dz
                                      = v 0 + 2gz
                                          2
                                                                             (3)
                                   dt
and after separation of variables and integration,
                                         h                       t
                                                     dz
                                         ∫
                                         0      v 0 + 2gz
                                                  2
                                                             = ∫ dt
                                                                 0

                                            h
                                        1       2g
                                   ⇔
                                        g0∫ 2 v 2 + 2gz dz = t
                                                0




                                   ⇔
                                     1
                                     g
                                            [v   2
                                                 0   + 2gz   ] =t
                                                             h

                                                             0



                                   ⇔
                                     1
                                     g
                                            [v   2
                                                 0   + 2gh − v 0 = t
                                                               2
                                                                         ]
and finally

                                   t=
                                        1
                                        g
                                            [v   2
                                                 0   + 2gh − v 0     ]       (4)



From equation (2) putting z = h , we obtain




                                                                                   85
                                v = v 0 + 2gh
                                      2
                                                                              (5)


which is the velocity the object hits the ground.




Using Mathematica, we may arrive at (4) by finding the general solution of the
                                                 dt        1
differential equation (3), written in the form      =
                                                 dz     v 0 + 2gz
                                                          2




and then determining the arbitrary constant C[1] , by t (0) = 0 .

We may also incorporate the initial condition in the code. In this case we find




1.2.3. Motion of a particle in Earth’s Gravitational Field.




                                                                                    86
                                                     r
The Earth is considered to be motionless. The force F acting on a particle of mass m at a
distance r from the center of the Earth is given by Newton’s law of Universal Gravity :
                                  r     mM r
                                  F = −G 2 k
                                         r
where
m : mass of the object
M : mass of the Earth.
                                                                   r
The origin of the axis is always Earth’s center and the unit vector k shows outward.


Example 1. An object is projected vertically upward from the earth’s surface with initial
speed v 0 . Neglecting air resistance

(a) find the speed at a distance H above the earth’s surface
(b) the smallest velocity of projection needed in order that the object never return (escape
    velocity).
Solution.
(a) In this case we combine Newton’s Second Law

                                              r
                                             dv r
                                           m    =F
                                             dt
and Newton’s Law of Universal Gravity :


                                           r     mM r
                                           F = −G 2 k
                                                  r


(Another useful relation is MG = gR 2 ).
We find
                                      dv r     mM r
                                  m      k = −G 2 k
                                      dt        r
                                        dv       1
                                  ⇔        = −GM 2                        (1)
                                        dt      r


From equation (1) we get


                                  dv dr       1
                                        = −GM 2
                                  dr dt      r


                                                                                               87
                                        dv         1
                                  ⇔        v = −GM 2                      (2)
                                        dr        r


and after separation of variables and integration,


                 v               R+H
                                       dr
                 ∫ vdv = −GM
                 v0
                                  ∫
                                  R    r2
                           v                 R +H
                    v2     1 
                 ⇔   = −GM  
                    2  v0  rR
                           2
                      v2 v0           1  1
                 ⇔      −    = −GM −    + 
                      2   2         R +H R
                                                H
                 ⇔ v 2 − v 0 = −2GM
                           2

                                            R (R + H )
                                 2GM   H
                 ⇔ v2 = v0 −
                         2
                                     ⋅                                    (3)
                                  R R+H


Since MG = gR 2 we get from (3) :


                                       2gRH
                          v2 = v0 −
                                2
                                                                          (4)
                                       R+H




Using Mathematica, we may arrive at (3) by finding the general solution of the differential
equation (2),




and then determining the arbitrary constant C[1] , by r ( v 0 ) = R .

We may also incorporate the initial condition in the code. In this case we find




                                                                                          88
(b) We suppose that the velocity of the object at an infinite distance (H → ∞ ) from earth’s

       surface is zero : v ∞ = 0

       and the object’s initial velocity is the escape velocity : v 0 = v esc .

       We then get from equation (4) :
                                                      H
                             0 = v esc − 2gR lim
                                   2
                                               H →∞ R + H


and since
                                              H
                                      lim        = 1,
                                      H →∞   R+H


we have finally

                             0 = v esc − 2gR ⇔ v esc = 2gR
                                   2
                                                                                  (5)




Example 2. An object is projected vertically upward from the earth’s surface
with initial speed v 0 .

(i)        Prove that the maximum height H reached above the earth’s surface is


                                                2
                                               v0R
                                      H=
                                             2gR − v 0
                                                     2




(ii)       Discuss the significance of the case where v 0 = 2gR .
                                                        2




                                                                                           89
                                                       2
                                                      v0
(iii)   Prove that if H is small, then it is equal to    very nearly.
                                                      2g
Solution.
Since the only force acting on the object is Newton’s force,


                                 r     mM r
                                 F = −G 2 k ,
                                        r
Newton’s Second Law
                                       r
                                      dv r
                                 m       =F
                                      dt
becomes
                                      dv r     mM r
                                 m       k = −G 2 k
                                      dt        r
or
                                  dv       1
                                     = −GM 2                            (1)
                                  dt      r
From eq (1) we get


                                  dv dr       1
                                        = −GM 2
                                  dr dt      r
                                       dv         1
                                 ⇔        v = −GM 2                     (2)
                                       dr        r


and after separation of variables and integration,




                 0              R+H
                                      dr
                 ∫ vdv = −GM ∫
                 v0              R    r2
                          0                   R +H
                    v2      1
                 ⇔   = −GM − 
                    2  v0   rR
                        2
                       v0         1  1
                 ⇔−       = −GM −   + 
                       2        R+H R
                        2
                       v0     1   1
                 ⇔        =−    +
                      2GM    R+H R




                                                                              90
                            2
                    1      v0   1
                 ⇔     =−     +
                   R+H    2GM R
                            2
                    1      v0     1
                 ⇔     =−       +
                   R+H    2gR 2
                                  R

                      1    − v 0 + 2gR
                               2
                 ⇔       =
                     R+H      2gR 2

                                2gR 2
                 ⇔ R+H=
                               2gR − v 0
                                       2



                           2gR 2
                 ⇔H=                −R
                          2gR − v 0
                                  2


                             2
                            v0 R
                 ⇔H=                                                      (3)
                          2gR − v 0
                                  2




Using Mathematica, we may arrive at (3) by finding the general solution of the differential
equation (2),




and then determining the arbitrary constant C[1] , by r ( v 0 ) = R .




We may also incorporate the initial condition in the code. In this case we find




We may also use the following




                                                                                          91
using from the beginning the fact that GM = gR 2 .
We further define the function r[ v] by




Equation (3) can be obtained if we solve the equation r[0] = H + R with
respect to H :




Example 3.     An object is projected vertically upward from the Earth’s surface with

initial speed v 0 . Prove that the time taken to reach the maximum height H is


                          R+H H R+H            R − H
                                 +    ⋅ arccos      
                           2g  R   2R          R + H




Solution.

Since the only force acting on the object is Newton’s force,

                                r     mM r
                                F = −G 2 k
                                       r
                       r
                      dv r
Newton’s Second Law m    =F
                      dt
becomes
                                    dv r     mM r
                                m      k = −G 2 k
                                    dt        r


                                                                                        92
or
                                    dv       1
                                       = −GM 2                        (1)
                                    dt      r

From equation (1) we get

                                    dv dr        1
                                          = −GM 2
                                    dr dt       r
                                       dv           1
                                    ⇔     v = −GM 2                   (2)
                                       dr          r

We first calculate the speed v of the object at a distance r from Earth’s center. We
separate variables in equation (2) and integrate :




                   v                  r
                                       dr
                   ∫ vdv = −GM ∫
                   v0                R r
                                         2

                         v                    r
                  v2       1
                ⇔   = −GM − 
                   2  v0   r R

                           2
                  v2 v0              1 1
                ⇔     −      = −GM − + 
                   2     2           r R
                                      1 1
                ⇔ v 2 − v 0 = −2gR 2  − + 
                          2

                                      r R
                                  2
                              2gR
                ⇔ v2 = v0 +
                          2
                                    − 2gR
                                r


                ⇔ v2 =
                         (v   2
                              0           )
                                  − 2gR r + 2gR 2
                                       r


                ⇔ v2 =
                                      (
                         2gR 2 − 2gR − v 0 r
                                         2
                                                  )                   (3)
                                  r




Using Mathematica, we may arrive at (3) by finding the general solution of the
differential equation (2),




                                                                                       93
and then determining the arbitrary constant C[1] , by r ( v 0 ) = R .

We may also incorporate the initial condition in the code. In this case we find




            dr
Since v =      , we get from equation (3)
            dt

                         dr 
                          =
                               2gR 2 − 2gR − v 0 r
                                2              2
                                                  (                  )
                         dt           r



from which we obtain


                         dr
                            =
                                             (
                                    2gR 2 − 2gR − v 0 r
                                                    2
                                                                 )               (3a)
                         dt                  r

which after separation of variables and integration, gives us

                        R +H                                              t
                                              r
                          ∫                                          dr = ∫ dt   (4)
                          R              (
                                2gR 2 − 2gR − v 0 r
                                                2
                                                             )            0



The previous equation can be further written as

                                        R+H
                                1                     r dr
                          2gR − v 0
                                  2      ∫
                                         R        a2 − r
                                                             =t                  (5)


where
                             2gR 2
                        a =
                          2
                                                                                 (6)
                            2gR − v 0
                                    2



It is obvious that since the maximum height H is

                                   2
                                  v0R
                        H=                                                       (7)
                                2gR − v 0
                                        2



(see previous Example),

                        a2 = R + H                                               (8)

Making the substitution r = u 2 in the integral (5), we have



                                                                                        94
                                             dr = 2udu

and for the new limits of integration we find

                                  r=R→u= R
                                  r =R+H→u = R+H

Integral (5) becomes

                                                   R+H
                                         1                  2u 2
                                      2gR − v 0
                                              2     ∫     a2 − u2
                                                                     du = t
                                                    R




               1                  1      R+H
or since                      =              (as follows from (6) and (8)),
           2gR − v 0
                   2              R       2g

                                          R +H
                   1         R+H                   2u 2
                   R          2g
                                 ⋅           ∫    a2 − u2
                                                            du = t                 (9)
                                             R


We also have

                             2u 2                    u
                   ∫       a2 − u2
                                   du = a 2 ⋅ arcsin   − u a 2 − u 2
                                                     a

and so

                       R+H                                                     R +H
                                  2u 2                  u               
                       ∫       a −u
                                2   2
                                      du = a 2 ⋅ arcsin   − u a 2 − u 2 
                                                        a               
                       R                                                       R




                     
                                  R+H                                 
                   = a 2 ⋅ arcsin            − R + H ⋅ a 2 − (R + H )  −
                                                                         
                                      a                               
                                                                      
                              
                                            R                 
                                                                 
                            − a 2 ⋅ arcsin     
                                             a − R ⋅ a −R
                                                          2

                              
                                                              
                                                                 

                                                                          
                   = {(R + H) ⋅ arcsin 1 − 0} − (R + H) ⋅ arcsin
                                                                   R
                                                                      − RH  =
                                                                 R+H      

                            π          R 
                   = (R + H) − arcsin     + RH                                   (10)
                            2         R+H



                                                                                          95
                 R
If arcsin           = ω then as it follows from next figure,
                R+H




π                               R
  − ω = φ and since φ = arccos     , we have
2                              R+H



                                π           R            R
                                  − arcsin     = arccos     .
                                2          R+H          R+H

So (10) gives us

            R +H
                    2u 2                                R
            ∫      a −u
                    2      2
                               du = (R + H) ⋅ arccos
                                                       R+H
                                                           + RH   (11)
             R




We finally get from (9)



                           1    R+H                       R 
                    t=               RH + (R + H) arccos    
                           R     2g                      R+H
or

                           R+H H R+H            R 
                    t=            +   ⋅ arccos                  (12)
                            2g  R   R          R+H

Using the identity
                                         R  1       R−H
                               arccos      = arccos
                                        R+H 2       R+H

(which we leave as an exercise) formula (12) can be written as

                               R+H  H R+H           R − H
                    t=                +    ⋅ arccos                    (13)
                                2g  R   2R          R + H




                                                                          96
Using Mathematica , we may solve differential equation (3a)




where C[1] is a constant determined from the initial conditions.


Example 4. Prove that if an object is dropped to the Earth’s surface from a height H, then
if air resistance is negligible it will hit the Earth with a speed

                                           2g R H
                                   v=
                                           R+H
Calculate also the time taken for the object to reach Earth’s surface.
Solution.




We start from the differential equation

                                   dv                1
                                         v = −GM                         (1)
                                    dr               r2

which we have derived in the previous examples.
Separation of variables and integration, we get


                           v               R
                                               dr
                          ∫ vdv = −GM
                           0
                                           ∫
                                          R +H r
                                                 2


                                     v                R
                            v2       1
                          ⇔   = −GM − 
                             2 0     r  R+H




                                                                                         97
                              v2       1   1 
                         ⇔       = −GM − +   
                              2        R R +H
                              v2            −H
                         ⇔       = −GM ⋅
                              2          R (R + H )
                                                   H
                         ⇔ v 2 = 2GM ⋅
                                               R (R + H)


                                                   H
                         ⇔ v 2 = 2GM ⋅
                                               R (R + H)
                                                   H
                         ⇔ v 2 = 2gR 2 ⋅
                                               R (R + H)

                                        2gRH
                         ⇔v=                               (2)
                                        R+H


We start again from eq . (1) and integrating,


                  v                 r
                                        dr
                  ∫ vdv = −GM
                  0
                                    ∫
                                   R +H r
                                          2


                          v                    r
                   v2       1
                 ⇔   = −GM − 
                    2 0     r  R+H

                       v2       1   1 
                 ⇔        = −GM − +   
                       2        r R +H
                            1   1 
                 ⇔ v 2 = 2GM −    
                            r R +H
                dr
and since v =      ,
                dt
                               2
                          dr    21   1 
                           = 2gR  −    
                          dt     r R +H


                         dr          R+H−r
                            = − 2g R
                         dt          r (R + H )


                         dr       2g   R+H−r
                            = −R     ⋅                     (3)
                         dt      R+H     r


                                                                 98
where we have included a minus sign. Separating variables and integration, we get from
eq.(3)


                               R+H
                                       R                       t
                        1                  r
                    −
                        R           ∫H R + H − r dr = ∫ dt
                                2g R +                0
                                                                        (4)



Under the substitution r = u 2 we have dr = 2udu and the new limits are


                    r =R+H→u = R+H
                    r=R→u= R
Eq. (4) becomes

                               R+H
                                            R
                      1                             2u 2
                    −
                      R         2g
                                   ⋅       ∫      R + H − u2
                                                               du = t           (5)
                                           R +H

We also have

                   2u 2                           
                       du = (R + H ) ⋅ arcsin
                                                u
         ∫       R+H−u         2
                                                    − u R + H − u2
                                              R+H 
and so
             R
                        2u 2
             ∫     R + H − u2
                                   du =
             R+H

                                                                          R
                                                          
                    = (R + H ) arcsin
                                         u
                                             − u R + H − u2                   =
                                      R+H                             R+H


                                               
                    = (R + H ) arcsin     − RH  − {(R + H ) arcsin(1) − 0}
                                        R
                                      R+H      

                    = (R + H ) arcsin               − RH − (R + H )
                                                 R                  π
                                                R+H                 2
                                            π
                    = (R + H ) arcsin
                                         R
                                            −  − RH =
                                      R + H 2

                                         R 
                    = (R + H ) − arccos     − RH =
                                        R+H

                       R + H        R−H      
                    = −      arccos     + RH  .
                        2           R+H      


                                                                                         99
We get finally from equation (5) :

                       1    R+H      R+H       R − H
                  t=             RH + 2 arccos R + H 
                       R     2g                      
or

                           R+H H R+H          R − H
                  t=              +    arccos      
                            2g  R   2R        R + H




1.2.4. Motion of a chain.

Example 1. A uniform chain of total length α has a portion b (0 < b < α ) hanging over the
edge of a smooth table AB. Prove that the time taken for the chain to slide off the table, if it
starts from rest, is

                               α      α + α2 − b2
                            t=   ⋅ ln             .
                               g          b
Solution.
Suppose that ρ is the linear density of the chain:




                                       m
                                  ρ=     .
                                       a
It is obvious that for a length ∆l and corresponding mass ∆m we also have



                                                                                              100
                                       ∆m
                                 ρ=       .
                                       ∆l
We suppose now that the chain moves a distance x. Then the mass which corresponds to
length x is m x = xρ . The total force which is responsible for the motion of the chain is
                                 r         r      r
                                 Fx = m x gk = xρgk
                       r
                      dv r
Newton’s Second Law m    = Fx              gives
                      dt
                                      dv r       r
                                 ρα      k = xρ gk
                                      dt
or
                                     dv
                                 α      = xg                              (1)
                                     dt
From eq (1) we get
                                     dv dx
                                 α         = xg ,
                                     dx dt
                                     dv
                                 α      v = xg
                                     dx
and after separation of variables and integration


                           0
                            v
                         a ∫ vdv = g ∫ xdx
                                           x
                                           b
                                                        (x > b )
                                      v                 x
                             v2   x2 
                         ⇔ a  = g 
                             2 0
                                  2 b
                                    

                                v2     x 2 b2 
                         ⇔a        = g    − 
                                2      2    2 
                                              
                                       (
                         ⇔ av2 = g x 2 − b 2        )
                                     g 2
                         ⇔v=           x − b2                                                (2)
                                     a
            dx
Since v =      we get from (2)
            dt
                                 dx            g
                                    =            x 2 − b2
                                 dt            a
and after separation of variables and integration




                                                                                             101
                          a                        t
                                dx             g
                          ∫
                          b   x 2 − b2
                                           =
                                               a   ∫ dt
                                                   0

                                                       a
                                                           g
                          ⇔ ln x + x 2 − b 2  =            t
                            
                                             b
                                                          a

                                 (                 )
                          ⇔ ln a + a 2 − b 2 − ln b =
                                                             g
                                                             a
                                                               t

and finally

                                        a      a + a 2 − b2
                                     t=   ⋅ ln
                                        g          b


Example 1a. – Chain Sliding Off Frictionless Table
Problem Description:

A uniform chain of total length ‘a’ is on a horizontal table so that a length ‘b’ dangles over
the side at time zero. Determine how long it will take the chain to slide off the table.
Define the density of the chain as σ (mass per unit length) and let x(t) be the length hanging
over the side at time t. Therefore, the force on the chain due to gravity is simply the weight of
the portion of the chain overhanging the table, or
                                     Fg = mg = σxg
But the total mass being accelerated is given by
                                               dv
                                     Ft = M       = σxg
                                               dt
And from Newton’s law we have
                                          dv
                                     σa      = σxg
                                          dt
Also, we know that position, x(t), and velocity, v(t), are related. In particular, one
relationship that is often useful is
                                     dv dv dx    dv
                                       =      =v
                                     dt dx dt    dx
Putting this into the above force balance gives the final mathematical model
                                         dv g
                                     v     = x
                                         dx a
with initial conditions x (0) = b and v(0) = 0 .
Solution Method:
1. The above equation is separable. Therefore, it can be written as




                                                                                             102
                                             g
                                   v dv =      x dx
                                             a
2. Integration gives

                           v2 g  x 2
                             = 
                           2 a 2
                                          
                                          
                                                    g
                                      + c  or v 2 = x 2 + c
                                                    a
                                                               (        )
                                         
or

                                            g 2
                                   v=         x +c
                                            a
3. Now applying the initial conditions gives

                                            g
                                   0=         b2 + c
                                            a

Therefore, c = − b
                     2
                         and

                                                  g 2
                                   v( x , t ) =     x − b2
                                                  a
This is the formal solution for v (x, t).
                                                      dx
4. To find x(t) explicitly, we note that v( t ) =        . Therefore,
                                                      dt

                                   dx   g
                                      =            x 2 − b2
                                   dt   a
Separating variables again gives

                                        dx             g
                                                  =      dt
                                      x 2 − b2         a
and, using standard integral tables, this becomes

                                          x   g
                                   cosh −1  =   ( t + c)
                                          b   a
5. To put this into a more usable form (there is no cosh −1 button on my calculator!), note
     that
                               1 x
                 cosh( x ) =     (e + e − x ) and cosh −1 (u ) = 1n ( u + u 2 − 1 )
                               2
Therefore,

                              x + x 2 − b2       
                           ln                     = g ( t + c)
                                  b                 a
                                                 


                                                                                         103
                                                                    b
6. Now applying the initial conditions again, we have that ln  = ln(1) = 0 .
                                                                    b
Therefore, c = 0 in the above expression. Finally, the time it takes for the chain to overhang
a distance x is given by

                               a  x + x 2 − b2             
                            T=  ln                         
                               g      b                    
                                                           
                                                       9 .8 m
7. For example, if a = 10m , b = 1m , and g =                 , then the time required for the chain
                                                         s2
    to fall off the table is simply (note that x = a for this case)

                           10 10 + 10 2 − 12 
                  T=          ln              = 1.01× ln(19.95) = 3.02 sec
                           9.8      1        
                                             
Thus, under these conditions, it takes about 3 sec for the chain to side completely off the
table.

Example 2. A uniform chain of total length a haw a portion b ( 0 < b < a ) hanging over
the edge of a smooth table. If the table has coefficient of friction µ prove that the time taken
for the chain to slide off the table if it starts from rest is

                                   a          a+         a 2 − [b (1 + µ) − aµ]2
                            t=           ⋅ ln
                               (1 + µ) g                  b (1 + µ) − a µ
Solution.




                                                                                                 104

				
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