USAPhO 2018 Problem A1

a. Suppose you drop a block of mass m vertically onto a fixed ramp with angle \theta with coefficient of static and kinetic friction \mu. The block is dropped in such a way that it does not rotate after colliding with the ramp. Throughout this problem, assume the time of the collision is negligible.
    i. Suppose the block’s speed just before it hits the ramp is v and the block slides down the ramp immediately after impact. What is the speed of the block right after the collision?
    ii. What is the minimum \mu such that the speed of the block right after the collision is 0 ?

b. Now suppose you drop a sphere with mass m, radius R and moment of inertia \beta m R^{2} vertically onto the same fixed ramp such that it reaches the ramp with speed v.
    i. Suppose the sphere immediately begins to roll without slipping. What is the new speed of the sphere in this case?
    ii. What is the minimum coefficient of friction such that the sphere rolls without slipping immediately after the collision?