# 7186062352 by panniuniu

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```									Momentum and Collisions
Objectives
 To compare the momentum of different
objects.
 To compare the momentum of the same
object moving with different velocities.
 To identify examples of change in the
momentum of an object.
 To describe changes in momentum in
terms of force and time.
Momentum
   The linear momentum pof an object of
mass m moving with a velocity is defined
as the product of the mass and the
velocity v
◦ p  mv
◦ SI Units are kg m / s
◦ Vector quantity, the direction of the
momentum is the same as the velocity’s
Momentum components
   p x  mv x and p y  mv y
   Applies to two-dimensional motion
 Momentum for any object at rest =
______________________
 Double the mass results in
________________ the momentum
 Quadruple the velocity results in
______________ the momentum.
   Determine the momentum of a …
◦ 60 kg halfback moving eastward at 9 m/s
◦ 1000 kg car moving northward at 20 m/s
◦ 40 kg freshman moving southward at 2 m/s
   A car possesses 20 000 units of
momentum. What would be the car’s
new momentum if …
◦ Its velocity were doubled
◦ Its velocity were tripled
◦ Its mass were doubled (by adding more
◦ Both its velocity and mass were doubled
Impulse
 In order to change the momentum of an object,
a force must be applied
 The rate of change of momentum of an object
is equal to the net force acting on it
p m(vf  vi )
◦                Fnet
t    t
◦ Gives an alternative statement of Newton’s second
law (F=ma, where a = (vf-vi)/∆t )
Impulse cont.
   When a single, constant force acts on the
object, there is an impulse delivered to
the object
◦ I  Ft
◦ I is defined as the impulse
◦ Vector quantity, the direction is the same as
the direction of the force
Impulse-Momentum Theorem
   The theorem states that the impulse
acting on the object is equal to the
change in momentum of the object
◦ Ft  p  mv f  mv i
◦ If the force is not constant, use the average
force applied
Average Force in Impulse
   The average force can be
thought of as the constant
force that would give the
same impulse to the
object in the time interval
as the actual time-varying
force gives in the interval
Average Force cont.
 The impulse imparted by a force during
the time interval Δt is equal to the area
under the force-time graph from the
beginning to the end of the time interval
 Or, the impulse is equal to the average
force multiplied by the time interval,
Fav t  p
Which of the above has the greatest velocity change?
The greatest acceleration?
The greatest momentum change?
The greatest impulse?
Which of the above has the greatest velocity change?
The greatest acceleration?
The greatest momentum change?
The greatest impulse?
A 1400 kg car moving westward with a velocity of 15
m/s collides with a utility pole and is brought to rest in
0.30 s.
   Find the magnitude of the force exerted
on the car during the collision.
A 2250 kg car traveling to the west slows down
uniformly from 20.0 m/s to 5.0 m/s.
   How long does it take the car to
decelerate if the force on the car is 8450
N to the east?

   How far does the car travel during the
deceleration?
   If the maximum coefficient of kinetic
friction between a 2300 kg car and road is
0.50, what is the minimum stopping
distance for a car moving at 29 m/s?
Impulse Applied to Auto Collisions
   The most important factor is the collision
time or the time it takes the person to
come to a rest
◦ This will reduce the chance of dying in a car
crash
   Ways to increase the time
◦ Seat belts
◦ Air bags
Air Bags
   The air bag increases the
time of the collision
   It will also absorb some of
the energy from the body
   It will spread out the area
of contact
◦ decreases the pressure
◦ helps prevent penetration
wounds
Review

 In order to change the momentum of an object
a ________ must be applied.
 How can the magnitude of an applied force be
reduced when an object undergoes a change in
momentum?
Conservation of Momentum
   Momentum in an isolated system in which a
collision occurs is conserved
◦ A collision may be the result of physical contact
between two objects
◦ “Contact” may also arise from the electrostatic
interactions of the electrons in the surface atoms of
the bodies
◦ An isolated system will have no external forces
Conservation of Momentum, cont

   The principle of conservation of
momentum states when no external forces
act on a system consisting of two objects
that collide with each other, the total
momentum of the system remains constant
in time
◦ Specifically, the total momentum before the
collision will equal the total momentum
after the collision
Conservation of Momentum, cont.
   Mathematically:
m1v1i  m2v2i  m1v1f  m2v2f
◦ Momentum is conserved for the system of objects
◦ The system includes all the objects interacting with
each other
◦ Assumes only internal forces are acting during the
collision
◦ Can be generalized to any number of objects
Think Pair Share
A boy standing at one end of a floating raft that is
stationary relative to the shore walks to the opposite
end of the raft, away from the shore. As a
consequence, the raft (a) remains stationary, (b)
moves away from the shore, or (c) moves toward the
shore. (Hint: Use conservation of momentum.)
   A 76 kg boater, initially at rest in a
stationary 45 kg boat, steps out of the
boat and onto the dock. If the boater
moves out of the boat with a velocity of
2.5 m/s to the right, what is the final
velocity of the boat?
   A baseball player wishes to maintain his physical
condition during the winter. He uses a 50.0 kg pitching
machine to help him, placing the machine on the
pitcher’s mound. The ground is covered with a thin
layer of ice so that there is negligible friction between
the ground and the machine. The machine fires a 0.15
kg baseball horizontally with a speed of 36 m/s. What is
the recoil speed of the machine?
Types of Collisions
 Momentum is conserved in any collision
 Inelastic collisions
◦ Kinetic energy is not conserved
 Some of the kinetic energy is converted into other types of
energy such as heat, sound, work to permanently deform an
object
 Example: rubber ball with hard floor
◦ Perfectly inelastic collisions occur when the objects
stick together
 Not all of the KE is necessarily lost
 Example: two pieces of putty, meteorite collides with the earth
More Types of Collisions
   Elastic collision
◦ both momentum and kinetic energy are
conserved
◦ Example: billiard balls, air molecules with the
walls of a container
   Actual collisions
◦ Most collisions fall between elastic and
perfectly inelastic collisions
Think Pair Share
A car and a large truck traveling at the same
speed collide head-on and stick together.
Which vehicle experiences the larger change in
the magnitude of its momentum? (a) the car
(b) the truck (c) the change in the magnitude
of momentum is the same for both (d)
impossible to determine
Collisions
 When two objects
stick together after
the collision, they have
undergone a perfectly
inelastic collision
 Conservation of
momentum becomes

m1v1i  m2v 2i  (m1  m2 )v f
Collisions
   Momentum is a vector quantity
◦ Direction is important
◦ Be sure to have the correct signs
An object of mass m moves to the right with a speed v.
It collides head-on with an object of mass 3m moving
with speed v/3 in the opposite direction. If the two
objects stick together, what is the speed of the
combined object, of mass 4m, after the collision?
(a) 0 (b) v/2 (c) v (d) 2v
A skater is using very low friction rollerblades. A friend
throws a Frisbee® at her, on the straight line along
which she is coasting. Describe each of the following
events as an elastic, an inelastic, or a perfectly inelastic
collision between the skater and the Frisbee: (a) She
catches the Frisbee and holds it. (b) She tries to catch
the Frisbee, but it bounces off her hands and falls to the
ground in front of her. (c) She catches the Frisbee and
immediately throws it back with the same speed
(relative to the ground) to her friend.
In a perfectly inelastic one-dimensional collision between
two objects, what condition alone is necessary so that
all of the original kinetic energy of the system is gone
after the collision? (a) The objects must have momenta
with the same magnitude but opposite directions. (b)
The objects must have the same mass. (c) The objects
must have the same velocity. (d) The objects must have
the same speed, with velocity vectors in opposite
directions.
An empty train car moving east at 21 m/s collides
with a loaded train car initially at rest that has
twice the mass of the empty car.
The two cars stick together.
A) Find the velocity of the cars after the collision.

B) Find the final speed if the loaded car moving at
17 m/s had hit the empty car initially at rest.
An empty train car moving at 15 m/s collides with
a loaded car of three times the mass moving at
one-third the speed of the empty car. The cars
stick together.
   Find the speed of the cars after the
collision.
A clay ball with a mass of 0.35 kg hits another 0.35
kg ball at rest, and the two stick together. The first
ball has an initial speed of 4.2 m/s.
   What is the final speed of the balls?

   Calculate the decrease in kinetic energy
that occurs during the collision.
   An 1800 kg car stopped at a traffic light is
struck from the rear by a 900kg car and
the two become entangled. If the smaller
car was moving at 20 m/s before the
collision, what is the speed of the
entangled cars after the collision?
   Two balls of mud collide head on in a
perfectly inelastic collision. Suppose m1 =
0.500 kg and m2 = 0.250 kg, v1i = 4 m/s
and v2i = -3.00 m/s. Find the velocity of
the composite ball of mud after the
collision. What is the change in kinetic
energy of the system after the collision?
 Both momentum and kinetic energy are
conserved
 Typically have two unknowns

m1v1i  m2 v 2i  m1v1f  m2 v 2 f
1        1         1        1
 Solve 1i  m2 v 2i  m1v1f  m2 2
m1vthe equations simultaneouslyv 2 f
2        2         2

2        2         2        2
Elastic Collisions, cont.
   A simpler equation can be used in place
of the KE equation

v1i  v 2i  (v1f  v 2f )
Two billiard balls each with a mass of 0.35 kg,
strike each other head on elastically.
   One ball is initially moving left at 4.1 m/s
and ends up moving right at 3.5 m/s. The
second ball is initially moving to the right
at 3.5 m/s. Find the final velocity of the
second ball.
Two nonrotating balls on a frictionless surface
 The first ball has a mass of 15 g and an initial
velocity of 3.5 m/s to the right, while the
second ball has a mass of 22 g and an initial
velocity of 4.0 m/s to the left. The final velocity
of the 15 g ball is 5.4 m/s to the left. What is
the final velocity of the 22 g ball?
Two billiard balls move toward one
another. The balls have identical masses
and assume that the collision between
them is elastic. If the initial velocities of
the balls are +30.0 cm/s and -20.0 cm/s,
what is the velocity of each ball after the
collision?
   Find the final velocity of the two balls if
the ball with initial velocity -20 cm/s has a
mass equal to half that of the ball with
initial velocity of +30 cm/s.
Summary of Types of Collisions
 In an elastic collision, both momentum and
kinetic energy are conserved
 In an inelastic collision, momentum is conserved
but kinetic energy is not
 In a perfectly inelastic collision, momentum is
conserved, kinetic energy is not, and the two
objects stick together after the collision, so
their final velocities are the same
Sketches for Collision Problems
 Draw “before” and
“after” sketches
 Label each object
◦ include the direction of
velocity
◦ keep track of subscripts
Sketches for Perfectly Inelastic
Collisions
 The objects stick
together
 Include all the velocity
directions
 The “after” collision
combines the masses
Glancing Collisions
   For a general collision of two objects in three-
dimensional space, the conservation of
momentum principle implies that the total
momentum of the system in each direction is
conserved
◦
m1v1ix  m2 v 2ix  m1v1f x  m2 v 2 f x and
Use  m2 v 2iy  m1v1f y  m v 2
◦m1v1iysubscripts for identifying 2thef yobject, initial and
final velocities, and components
Glancing Collisions

   The “after” velocities have x and y components
   Momentum is conserved in the x direction and in the y
direction
   Apply conservation of momentum separately to each
direction
Problem Solving for Two-
Dimensional Collisions
   Coordinates: Set up coordinate axes
and define your velocities with respect to
these axes
◦ It is convenient to choose the x- or y- axis to
coincide with one of the initial velocities
   Draw: In your sketch, draw and label all
the velocities and masses
Problem Solving for Two-
Dimensional Collisions, 2
 Conservation of Momentum: Write
expressions for the x and y components of the
momentum of each object before and after the
collision
 Write expressions for the total momentum
before and after the collision in the x-direction
and in the y-direction
Problem Solving for Two-
Dimensional Collisions, 3
   Conservation of Energy: If the
collision is elastic, write an expression for
the total energy before and after the
collision
◦ Equate the two expressions
◦ Fill in the known values
 Can’t be simplified
Problem Solving for Two-
Dimensional Collisions, 4
   Solve for the unknown quantities
◦ Solve the equations simultaneously
◦ There will be two equations for inelastic
collisions
◦ There will be three equations for elastic
collisions
A billiard ball moving with speed 3.0 m/s in the +x
direction strikes an equal mass ball initially at rest .
The two balls are observed to move off at 45o,
ball 1 above the x axis and ball 2 below. What
are the speeds of the two balls?
Lab Momentum, Energy and
Collisions
 Individually answer the preliminary questions on
a sheet of paper.
 When completed get my initials on your paper.
 Staple this to your lab report when you turn it
in.
 You will each type and turn in your data table
and analysis questions with prelim questions
attached.
.

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