Finding the Maximum Mass Ratio for Elastic Collisions: A Quick Homework Problem

[PeppaPig]

Homework Statement

Both object A and B have mass M and are moving.
An object with mass of m and moving with velocity u collide with the object B elastically. (m < M)
Find the following after m collide with B
1) Velocity of m after bouncing back
2) Velocity of B
Then m collide with A and bounce back
3) The highest value of M/m that velocity of m is lesser than or equal to that of B
(Surface has no friction)

Homework Equations

The Attempt at a Solution

Using the Law of Energy Conservation and the Law of Momentum Conservation

$\frac{1}{2} m u^2 = \frac{1}{2} m v_1^2 + \frac{1}{2} M v_2^2$

$m u = -m v_1 + Mv_2$

After solving the equation, I get this

$v_1 = \frac{M - m}{M + m} u$ (Answer for (1))

$v_2 = \frac{2 m}{M + m} u$ (Answer for (2))

Then object m hit object A. The velocity of m then change into

$(\frac{M - m}{m + M})^2 u$

Velocity of m should be equal to that of B so m cannot reach B

$(\frac{M - m}{m + M})^2 u = \frac{2 m u}{m + M}$

$\frac{M}{m} = 2 + \sqrt{5}$ (Answer for (3))

Is that correct?

 

[kuruman]

Looks fine.