A tennis ball of mass 49.0 g is held just above a basketball of mass 596 g. With

Question

A tennis ball of mass 49.0 g is held just above a basketball of mass 596 g. With their centers vertically aligned, both are released from rest at the same moment, to fall through a distance of 1.36 m, as shown in the figure below.
The magnitude of the downward velocity with which the basketball reaches the ground is 5.16 m/s in which we assume an elastic collision with the ground instantaneously reverses the velocity of the basketball while the tennis ball is still moving down.

QUESTION: the two balls meet in an elastic collision. To what height does the tennis ball rebound?

Explanation / Answer

Because this is an elastic collision, we must use both the conservation of momentum and the conservation of energy to solve the problem. (Note the first answer is incorrect as it incorrectly assumes that the after collision velocity of the basketball is negligible.) So, let
u1 = before collision velocity of the tennis ball = -5.16 m/s
u2 = before collision velocity of the basketball = 5.16 m/s
v1 = after collision velocity of the tennis ball
v2 = after collision velocity of the basketball
m1 = mass of the tennis ball = 49 g
m2 = mass of the basketball = 596 g

Then conservation of energy says
m1u1^2/2 + m2u2^2/2 = m1v1^2/2 + m2v2^2/2

and conservation of momentum says
m1u1 + m2u2 = m1v1 + m2v2

The only two true unknowns in these two equations are v1 and v2 and with two equations you can solve them.

So the solution for v1 is
v1 = (u1(m1 - m2) + 2m2u2) / (m1 + m2)
v1 = ((-5.16 m/s)(49 g - 596 g) + 2(596 g)(5.16 m/s)) / (49 g + 56 g)
v1 = 13.912 m/s

Anyway, now we have an upwards velocity for the tennis ball. The kinetic energy for for this will be converted entirely into potential energy when the tennis ball reaches maximum height, so
KE = (1/2)m1*v1^2 = PE = F*d = m1*g*d
Where
KE = Kinetic Energy of the tennis ball immediately after the collision
PE = Potential Energy of the tennis ball at the peak of it's bounce up
g = acceleration due to gravity
d = height the ball is at when it reaches the peak
So
(1/2)m1*v1^2 = m1*g*d
(1/2)*v1^2 = g*d
d = v1^2/(2*g)
d = (13.912 m/s)^2/(2 * 9.807 m/s^2)
d = 9.874 m

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