A small block of mass M is released from rest at the top of the curved frictionl

Question

A small block of mass M is released from rest at the top of the curved frictionless ramp shown above. The block slides down the ramp and is moving with a speed 3.5v_0 when it collides with a larger block of mass 1.5M at rest at the bottom of the incline. The larger block moves to the right at a speed 2v_0 immediately after the collision. Express your answers to the following questions in terms of the given quantities and fundamental constants. Determine the height h of the ramp from which the small block was released. Determine the speed of the small block after the collision. The larger block slides a distance D before coming to rest. Determine the value of the coefficient of kinetic friction Mew between the larger block and the surface on which it slides. Indicate whether the collision between the two blocks is elastic or inelastic. Justify your answer.

Explanation / Answer

a)

At the bottom of the frictionless ramp,
v = (2gh) = 3.5Vo square both sides
2gh = 12.25(Vo)²
h = 6.125(Vo)² / g

b)

Conserve momentum: M*3.5Vo + 0 = M*v + 1.5M*2Vo
M cancels, leaving 3.5Vo = v + 3Vo
Then v = 0.5*Vo

c)

initial KE = work done by friction
½(1.5M)(2Vo)² = µ(1.5M)gD M cancels, leaving
3*(Vo)² = 1.5µgD
µ = 2(Vo)² / gD

d)

I can tell by observation that the collision is inelastic (not completely, but inelastic). The reason is that in an elastic collision, when a small mass collides with a larger mass at rest, the smaller mass must rebound in the opposite direction. That didn't happen here.
A quantitative justification lies in testing conservation of energy:
initial KE = ½ * M * (3.5Vo)² = 6.125M(Vo)²
final KE = ½ * M * (0.5Vo)² + ½ * (1.5M) * (2*Vo)² = 3.125M(Vo)²

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