A block of mass m slides along a frictionless track with speed vm. It collides with a stationary block of mass M. Find an expression for the minimum value of vm that will allow the second block to circle the loop-the-loop without falling off if the collision is (a) perfectly inelastic or (b) perfectly elastic.

Following are the calculations to the perfectly inelastic and perfectly elastic.

For point a) perfectly inelastic:

Since we all know, collisions are totally inelastic:

In order to complete the circle:

For point b) perfectly elastic:

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Part a) When collision is perfectly inelastic

Part b) When collision is perfectly elastic

Explanation:

Part a)

As we know that collision is perfectly inelastic

so here we will have

so we have

now we know that in order to complete the circle we will have

now we have

Part b)

Now we know that collision is perfectly elastic

so we will have

now we have