Elastic collision between ball and block

1. The problem statement, all variables and given/known data
A block with large mass M slides with speed V0 on a frictionless table towards a wall. It collides elastically with a ball with small mass m, which is initially at rest at a distance L from the wall. The ball slides towards the wall, bounces elastically, and then proceeds to bounce back and forth between the block and the wall.

a) How close does the block come to the wall ?
b) How many times does the ball bounces off the block, by the time the block makes its closest approach to the wall?
Assume that M >> m, and give your answers to leading order in m/M.

2. Relevant equations

3. The attempt at a solution

Let the velocity of block and ball be v1 and v2 after collision. Applying conservation of momentum and law of restitution,

[itex]v_1 = \dfrac{M-m}{M+m} v_0 \\
v_2 = \dfrac{2M}{M+m} v_0 [/itex]

Using approximations
[itex]v_1 = v_0 \\ v_2 = 2v_0 [/itex]

At distance of closest approach the velocity of bigger block should become zero. But I can’t think how to find that distance as the velocity of bigger block is not constant and changes after every collision.


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