AP Physics Conservation of Momentum

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AP Physics  Momentum AP Wrapup
There are two, and only two, equations that you get to play with:


          p  mv

                 This is the equation for momentum.

          J  F t  p

                 This is the equation for impulse. The equation sheet uses, for some reason, the
                 symbol J for impulse (the Physics Kahuna has never seen this anywhere else. Oh
                 well.)

Here is what you are supposed to be able to do.

A. Systems of Particles, Linear Momentum

     1.     Impulse and Momentum: You should understand impulse and linear momentum so you
           can:

          a. Relate mass, velocity, and linear momentum for a moving body, and calculate the total
             linear momentum of a system of bodies.

          Just use the good old momentum equation.

          b. Relate impulse to the change in linear momentum and the average force acting on a
             body.

          Just use the impulse equation. We did several of these problems. If you have a collision,
          time, and a force in a problem, think' impulse'.

     2. Conservation of Linear Momentum, Collisions

          a. You should understand linear momentum conservation so you can:

             (1) Identify situations in which linear momentum, or a component of the linear
                 momentum, is conserved.

             The momentum of an isolated system (no outside forces) is conserved in any
             interaction. Period.

             (2) Apply linear momentum conservation to determine the final velocity when two
                 bodies that are moving along the same line, or at right angles, collide and stick
                 together, and calculate how much kinetic energy is lost in such a situation.



                                                                                                     147
             This is the good old inelastic collision problem. We've done a bunch. If the collision
             happens at right angles, then you are going to have to look at components of
             momentum (which you can do since momentum is a vector, right?) Anyway, the idea
             is that after the collision both bodies have the same velocity. Therefore:

                 m1v1  m2v2   m1  m2  v f

             If the two bodies are moving at right angles, then it's a bit more complicated. The
             vertical momentum is conserved and the horizontal momentum is conserved, so you
             can write equations for the conservation of momentum in the x and y directions. Then
             use these to solve for whatever unknown you've been presented with.

             (3) Analyze collisions of particles in one or two dimensions to determine unknown
                 masses or velocities, and calculate how much kinetic energy is lost in a collision.

             Pretty much all the problems that you will see will involve finding one unknown
             velocity. Body A moving at velocity v has an eleastic collision with body A which is at
             rest. Body B ends up with a velocity of vf, &tc. That sort of thing.

Let's look at a few test problems. The first one we'll look at is off 2001 test.




     An incident ball A of mass 0.10 kg is sliding at 1.4 m/s on the horizontal tabletop of negligible
     friction shown above. It makes a head-on collision with a target ball B of mass 0.50 kg at rest at
     the edge of the table. As a result of the collision, the incident ball rebounds, sliding backwards
     at 0.70 m/s immediately after the collision.

   (a) Calculate the speed of the 0.50 kg target ball immediately after the collision.

We use conservation of momentum to solve this one.
                                                                    m1  v1  v1 '
 m1v1  m1v1 ' m2v2 '           m1v1  m1v1 '  m2v2 '         v2 ' 
                                                                         m2
              m        m
      1.0 kg 1.4   0.70  
                 s     s                                   m
v2 '                                                0.42
               0.50 kg                                     s


                                                                                                       148
The tabletop is 1.20 m above a level, horizontal floor. The target ball is projected horizontally and
initially strikes the floor at a horizontal displacement d from the point of collision.

 (b) Calculate the horizontal displacement d.

This is a projectile motion problem. We know the ball's horizontal velocity and the height of the
table, so we can easily find the horizontal distance it travels as it falls.

                   1 2             2y           2 1.2 m 
Time to fall: y      at     t                                  0.495 s
                   2               a                 m
                                                  9.8 2
                                                     s
The ball has a horizontal velocity of 0.42 m/s (which we just figured out), so the distance d is
simply:

                     m
x  vt         0.42      0.495 s                 0.208 m
                     s




In another experiment on the same table, the target ball B is replaced by target ball C of mass 0.10
kg. The incident ball A again slides at 1.4 m/s, as shown above left, but this time makes a glancing
collision with the target ball C that is at rest at the edge of the table. The target ball C strikes the
floor at point P, which is at a horizontal displacement of 0.15 m from the point of the collision, and
at a horizontal angle of 30 from the + x-axis, as shown above right.

(c) Calculate the speed v of the target ball C immediately after the collision.

This sounds very hard, angles and all that stuff, right? Except we know the distance it traveled
(1.5 m) and we know how long it is in the air before it hits  same as the previous problem time.
So this is a ridiculously simple problem. Using the horizontal distance and the time to fall we can
find the horizontal velocity, which is the velocity it began with, which is the velocity right after the
collsion.

               x       0.15 m                    m
Find v:   v                               0.30
               t       0.495 s                   s



                                                                                                    149
(d) Calculate the y-component of incident ball A's momentum immediately after the collision.

We know that momentum is conserved in the x and y directions. So we can sum momentum in
the y direction. We know that this momentum has to add up to be zero as the ball A had no initial
momentum in the y direction.

                                                                     mcvc  sin 
0  mavay  mcvcy          mavay  mcvc  sin                   vay 
                                                                        ma

                        m
       0.10 kg    0.30  sin 30o
                        s
                                                                 m
vay                                                       0.15
                0.10 kg                                          s

From 1996:

   Two identical objects A and B of mass M move on a one-dimensional, horizontal air track.
   Object B initially moves to the right with speed vo. Object A initially moves to the right with
   speed 3vo, so that it collides with object B. Friction is negligible. Express your answers to the
   following in terms of M and vo.




     a.    Determine the total momentum of the system of the two objects.

We simply add up the momentum of each object to get the total momentum.

                 ptot  mAv A  mBvB  M  3v0   Mv0                       4Mv0

     b.    A student predicts that the collision will be totally inelastic (the objects stick together on
           collision). Assuming this is true, determine the following for the two objects
           immediately after the collision.

           i.   The speed.

This is an inelastic collision. It is a simple thing to solve:

                                                           mAv A  mBvB
          mAv A  mBvB   mA  mB  v f                  vf 
                                                             mA  mB 



                                                                                                      150
            M  3 v0   Mv0          4Mv0
     vf                                             2v0
              M  M                  2M

      ii.    The direction of motion (left or right).
             The combined object moves right. All values are positive.
When the experiment is performed, the student is surprised to observe that the objects separate
after the collision and that object B subsequently moves to the right with a speed 2.5 v0 .

c.    Determine the following for object A immediately after the collision.
      i.   The speed of object A after the collision.
      mAvA  mBvB  mAvA ' mBvB '                   mAvA  mBvB  mBvB '  mAvA '

              mAv A  mBvB  mBvB '              M  3 v0   Mv0  M  2.5 v0 
      vA '                                  
                      mA                                  M

      vA '   3 v0   v0   2.5 v0                   1.5 v0

      ii.    The direction of motion (left or right).
             Object A move right. All values are positive.
d.    Determine the kinetic energy dissipated in the actual experiment.

The kinetic energy dissipated is the change in kinetic energy. So we solve this question by
finding the kinetic energy before the collision and the kinetic energy after the collision. The
difference in the two is the energy dissipated.

    1         1                                      1           1
K i  m  v A   m  vB                              K f  m  v A '   m  vB ' 
            2        2                                         2          2
    2         2                                      2           2

   1         1                                     1           1
Ki  M  3vo   M  vo                             K f  M  2.5vo   M 1.5vo 
           2        2                                        2          2
   2         2                                     2           2


Ki  5Mvo 2                        K f  4.25Mvo 2


K dissipated  Ki  K f         5Mvo 2  4.25Mvo 2              0.75Mvo 2


                                                                                           151
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