Predicting Final Velocities for a Head-On Elastic Collision Problem

[~Fluffy]

Homework Statement

"Consider a head-on, elastic collision between two modies whose masses are m and M, with m << M. It is well known that if m has speed v_0 and M is initially t rest, m will bounce straight back with its speed unchanged, while M will remain at rest (to an excellent approximation). Use this fact to predict the final velocities if M approaches with speed v_0 and m is initially at rest. [Hint: Consider the reference frame attached to M.]

Homework Equations

1. p1 + p2' = p1 + p2'
2. m1*v1 + m2*v2 = m1*v1' + m2*v2' (elastic)
3. m1*v1 + m2*v2 = v' * (m1+m2) (inelastic)

The Attempt at a Solution

I came up with three possible scenarios for the answer. I'm not certain how to consider the reference frame hint into the problem, which is what I'm primarily interested in understanding.

My possible solutions:

1) M bounces off m, similar to how the example in the beginning of the problem.

ViM = V_0, Vim = 0.

VfM = V_0, Vfm = 0.

I don't think that's likely.

2) M stops at collision, all momentum is transferred to m.

ViM = V_0, Vim = 0.

VfM = 0, Vfm = (M / m) * V_0

3) Both m and M move at speed V_0 (roughly). I'm assuming that M is pushing m forward (so same speed).

ViM = V_0, Vim = 0.

VfM = Vfm = (M / (M+m)) * V_0 (rearrangement of equation 3)

My intuition tells me that solution 3 makes the most sense, but I would like to confirm its validity, as well as better understand the reference frame hint in relation to this problem.

Thank you! :)

 

[voko]

In the reference frame attached to M, what is the velocity of M? Of m?

 

[mruss]

Disclaimer: new poster to this forum so let me know if this is giving too much help

M is moving at velocity of v_0 in a static reference frame. But from the reference frame of M, it is not moving at all (almost by definition). Instead from M's point of view, it looks like m is approaching M at speed v_0 and M is standing still.

So if m was approaching M with a speed of v_0 the problem explained what would happen; it would bounce off and move away from M with a speed of v_0.

Does this give you a sense of how to use M's reference frame to answer the question?