Some easy impulse/momentum questions?

Question

1. 105 kg running back runs out of bounds at 8 m/s and collides with a 50 kg cheerleader at rest. If the final velocity of the running back is 5 m/s, what is the magnitude of the resulting velocity of the cheerleader after the collision?

2. A rollercoaster car is coasting at 7 m/s when it collides with 4 rollercoaster cars at rest. They couple after colliding. What is the magnitude of their resulting velocity? Name whether it is going forward or backward as well.

3. Two billiard balls of equal mass are on a billiard table. One ball is set in motion so that it strikes the second ball off center. The second ball travels 53 degrees to the left with a velocity of 6 m/s while the first ball travels 37 degrees to the right with a velocity of 8 m/s. Determine the magnitude of the velocity of the first ball prior to the collision.

The reason I don't know how to do these is because I don't know the equation!!! Thanks!

Answer 1

Conservation of momentum or
initial momentum = final momentum

where momentum is a vector quantity given by Mass x Velocity.

Looking at the last problem as it contains all the ingredients, let v0 be the initial velocity and let's pretend it is along the positive X axis.

Since the two balls have the same mass, we can just call it M. Since the initial velocity of the second ball is 0, we have:

P0 = M + <0, 0> M = M (or v0M in the positive X direction, 0 in the Y)

Therefor, since we must end with the same final momentum, the sum of the X components of the velocities of the two balls must be v0, while the sum of the Y components of the two velocities must be 0 (recall that one ball has a positive Y component while the other has a negative Y component).

Since you know the final velocities and angles of the two balls, you can compute the X and Y components. Then the answer is just the sum of the X components.

(This is somewhat of a trick question because you shouldn't really need both angles and both final velocities.)