# Conservation of Momentum/Mechanical Energy Question (Fairly Challenging)

## Homework Statement

A small block of mass m is moving on a horizontal table surface at initial speed v0. It then moves smoothly onto a sloped big block of mass M. The big block can also move on the table surface. Assume that everything moves without friction

Find the speed v of the small block after it leaves the slope.

## Homework Equations

Conservation of Momentum: mv0 = (M+m)v → v=(mv0)/(M+m)
Conservation of Mechanical Energy: fairly obvious
The height that it rises to is h = (1/2g)(Mv0^2/M+m), I derived this from conservation of momentum and conservation of mechanical energy

## The Attempt at a Solution

I set up the conservation of momentum: (M + m)v = -mv1 + Mv2 Is this correct?
Now I ask, how should I set up the conservation of mechanical energy equation? Should I solve for one variable in terms of the other? The answer should be in terms of v0.
The answer is v1 = ((M-m)/(M+m))v0

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Doc Al
Mentor
Conservation of Momentum: mv0 = (M+m)v → v=(mv0)/(M+m)
Don't assume that m & M have the same velocity.
Conservation of Mechanical Energy: fairly obvious
The height that it rises to is h = (1/2g)(Mv0^2/M+m), I derived this from conservation of momentum and conservation of mechanical energy
Don't worry about the height. That's an intermediate point that we don't care about.

## The Attempt at a Solution

I set up the conservation of momentum: (M + m)v = -mv1 + Mv2 Is this correct?
Set initial momentum equal to final momentum. Let v1 and v2 be the final velocities after they separate.
Now I ask, how should I set up the conservation of mechanical energy equation?
Intial KE = final KE
Should I solve for one variable in terms of the other? The answer should be in terms of v0.
You'll have two equations. Eliminate v2 and solve for v1.

How would a conservation of kinetic energy equation look like?

1/2(M+m)v^2 = 1/2mv1^2 + 1/2Mv2^2 ?

Conservation of momentum: (M+m)v = -mv1 + Mv2 ?

If I solve for v1 using these equations, it turns out very convoluted and does not resemble the answer. I feel as though it should be cleaner than this. Am I doing something incorrectly?

Does this problem even meet the criteria to use conservation of kinetic energy? It doesn't seem to be elastic, does it?

Doc Al
Mentor
How would a conservation of kinetic energy equation look like?

1/2(M+m)v^2 = 1/2mv1^2 + 1/2Mv2^2 ?
OK, except for the left hand side. Initially, only m is moving.

Conservation of momentum: (M+m)v = -mv1 + Mv2 ?
Same issue as above.

If I solve for v1 using these equations, it turns out very convoluted and does not resemble the answer. I feel as though it should be cleaner than this. Am I doing something incorrectly?
Clean up the equations as I suggest and try again. A bit of a pain, but you'll get the required answer with a bit of work.

Wow, it worked! Thanks so much for your help. I greatly appreciate it.

Doc Al
Mentor
Excellent! (And you are most welcome.)