You drop a small ball on top of a large ball from rest at a height, h (see figur

Question

You drop a small ball on top of a large ball from rest at a height, h (see figure below).

This is quite a dramatic collision. You drop a small ball on top of a large ball from rest at a height, h (see figure below). In this question you will calculate how high the small ball will bounce up. Assume al collisions are elastic so that you can use (vA- VB)-(vA- VB) a) What is the velocity of the small ball just when the large ball hits the floor? mB b) What is the velocity of the large ball just after it leaves the floor? c) Now the balls collide (after the large ball hits the floor). Use the conservation of momentum and that the collision is elastic to solve for the velocity of the small ball after the two balls collide. Assume that the mass of the large ball is much greater than the mass of the small ball so that you can ignore all terms involving the mass of the small ball.

Explanation / Answer

(A) mA g h = mA v^2 / 2

vA = - sqrt(2 g h)

(B) vB = sqrt(2 g h )

(c) Applying momentum conservation,

mA vA + mB vB = mA vAf + mB vBf

for elastic collision

vB - vA = vAf - vBf

vAf = vBf + 2 sqrt(2 g h )

putting in previous eqtn,

sqrt(2 g h ) (mB - mA) = mA vB f + 2 mA sqrt(2 g h) + mB vBf

sqrt(2 g h )(mB - 3 mA) = (mA + mB ) vBf

vBf = sqrt(2 g h )(mB - 3 mA) / (mA + mB )

vAf = sqrt(2 g h) (3mB - mA ) / (mA + mB )

(d) hA = vAf^2 / 2 g

= h [ (3mB - mA ) / (mA + mB ) ]^2

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