Physics 1A Introduction to Physics and Problem Solving

							      Lecture 4:
Vectors & Components
             Questions of Yesterday
1) A skydiver jumps out of a hovering helicopter and a few
   seconds later a second skydiver jumps out so they both
   fall along the same vertical line relative to the helicopter.
1a) Does the difference in their velocities:
   a) increase
   b) decrease
   c) stay the same
1b) What about the vertical distance between them?

2) I drop ball A and it hits the ground at t1. I throw ball B
    horizontally (v0y = 0) and it hits the ground at t2. Which is
    correct?
    a) t1 < t2
    b) t1 > t2
    c) t1 = t2
        Vector vs. Scalar Quantities
     Vector Quantities: Magnitude and Direction
      Ex. Displacement, Velocity, Acceleration

               Scalar Quantities: Magnitude
             Ex. Speed, Distance, Time, Mass

                                  What about 2 Dimensions?
     Vectors in 1
      Dimension                             3        y (m)
  Direction specified                       2
   solely by + or -                         1
                                                                 x (m)
                          x (m)     -3   -2 -1   1    2      3
                                            -1
-3 -2 -1 0    1   2   3                     -2
                                           -3
     Vectors: Graphical Representation
         Vector Quantities: Magnitude and Direction
                Represent in 2D with arrow
            Length of arrow = vector magnitude
             Angle of arrow = vector direction

           y (m)
                         R m at qo above x-axis
          3
          2                               Position of vector
          1                                 not important
     q              q        x (m)         Vectors of equal
-3 -2 -1           1q2   3           length & direction are equal
       -1
         -2
                                      Can translate vectors for
              q                       convenience (choose ref
         -3                                    frame)
  Adding Vectors: Head-to-Tail
Must have same UNITS (true for scalars also)
 Must add magnitudes AND directions..how?

 A+B=?
                      Head-to-Tail Method



                          A+B
Adding Vectors: Commutative Property

          A+B=B+A?


A+B                              B+A



                YES!
          A+B=B+A
        Can add vectors in any
                order
      Subtracting Vectors
             A -> -A
Negative of vector = 180o rotation

                     A - B = A + (-
                           B)



                           A-B
Multiplying & Dividing Vectors by Scalars
  2 * A = 2A             -2 * A = -2A




  Ex. v = x/t
   t=3s
            Graphical Vector Techniques
                                                            N
                             1 box = 10 km
  A plane flies from base
                                                      W            E
camp to lake A a distance
280 km at a direction 20o                  lake B           S
   north of east. After
dropping off supplies, the
   plane flies to lake B,
   which is 190 km and
 30.0o west of north from                           30o
          lake A.
 Graphically determine                                    lake A
    the distance and
direction from lake B to
     the base camp.
                                     20o
                              base camp
                 Vector Components
       Every vector can be described by its
                     components
  Component = projection of vector on x- or y-
   y                       y
                     axis

                     B                          R
                                     q          y
                         x                          x
             A                           R
                                         x
From magnitude (R) and direction   Rx = Rcosq
(q) of R can determine Rx and Ry   Ry = Rsinq
              Vector Components
Can determine any vector
  from its components      y

    R2 = Rx2 + Ry2
   R = (Rx2 + Ry2)1/2                  R
                               q       y
      tanq = Ry/Rx                         x
    q = tan-1(Ry/Rx)               R
     -90 < q < 90                  x
              Vector Components
Can determine any vector          Careful!
  from its components                    y

    R2 = Rx2 + Ry2          (-x, +y)         (+x, +y)
   R = (Rx2 + Ry2)1/2
                                    II       I
                                                          x
      tanq = Ry/Rx                 III       IV
    q = tan-1(Ry/Rx)         (-x, -y)            (+x, -
     -90 < q < 90                                  y)

                 I, IV: q = tan-1(Ry/Rx)
          II, III: q = tan-1(Ry/Rx) + 180o
      Important to know direction of vector!
      Vector Addition: Components
        Why are components useful?

  When is magnitude of A + B = A + B ?

              A           B

                    A+B
                          R x = A x + Bx +
                                Cx….
R = A + B + C…. =
                          R y = A y + By +
        ?
                                Cy….
                          q = tan-1(Ry/Rx)
         Vector Addition: Components


                                      lake B
  Using components
determine the distance
and direction from lake
 B to the base camp.                           30o

                                                 +
                                 Rx = Ax + Bx lake A
                                        Cx….
                                  R y = A y + By +
                                20o     Cy….
                          base camp
                                 q = tan-1(Ry/Rx)
                                  -90 < q < 90
       Vector Components: Problem #2
  A man pushing a mop across a floor cause the mop to
undergo two displacements. The first has a magnitude of
 150 cm and makes an angle of 120o with the positive x-
axis. The resultant displacement has a magnitude of 140
cm and is directed at an angle of 35.0o to the positive x-
  axis. Find the magnitude and direction of the second
                      displacement.
       Vector Components: Problem #3
 An airplane starting from airport A flies 300 km east,
   then 350 km at 30.0o west of north, and then 150 km
     north to arrive finally at airport B. The next day,
    another plane flies directly from A to B in a straight
                             line.
a) In what direction should the pilot travel in this direct
                            flight?
      b) How far will the pilot travel in the flight?
              Questions of the Day
1) Can a vector A have a component greater than its
   magnitude A?
a) YES
b) NO

2) What are the signs of the x- and y-components
of A + B in this figure?
   a) (x,y) = (+,+)
   b) (+,-)
   c) (-,+)
   d) (-,-)