A tennis ball of mass 58.0 g is held just above a basketball of mass 592 g. With

Question

A tennis ball of mass 58.0 g is held just above a basketball of mass 592 g. With their centers vertically aligned, both are released from rest at the same moment, to fall through a distance of 1.02 m, as shown in the figure below.

(a) Find the magnitude of the downward velocity with which the basketball reaches the ground. Assume an elastic collision with the ground instantaneously reverses the velocity of the basketball while the tennis ball is still moving down.

(b) Next, the two balls meet in an elastic collision. To what height does the tennis ball rebound?

Explanation / Answer

(a) The balls undergo free fall under the influence of gravity.

For basketball, u = 0, h = 1.02 m, g = 9.81 m/s2, v =?

v2 - u2 = 2gh

i.e. v2 - 0 = 2 x 9.81 x 1.02

or vbasketball = 4.47 m/s (on reaching the ground)

After undergoing elestic collision with the ground, the ball reverses its velocity, i.e. it becomes 4.47 m/s in upward direction.

(b) Having fallen throught the same altitude, the tennis ball too has the same velocity of 4.47 m/s in the downward direction just when it undergoes an elastic collision with the basketball.

Conserving momentum,

mbasketball,i ubasketball,i + mtennis,i utennisl,i = mbasketball,i ubasketball,f + mtennis,i utennis,f

Taking upward direction to be positive and downward direction to be negative, we have just before and after the collision:

(592 x 10-3 x 4.47) + (58 x 10-3 x (-4.47)) = (592 x 10-3 x ubasketball,f) + (58 x 10-3 x utennis,f)

or, 2387 = 592 ubasketball,f + 58 utennis,f ------- (1)

Also, for elastic collision, coefficient of restitution (e) = 1, i.e.

(ubasketball,f - utennis,f) = - 1 x (ubasketball,i - utennis,i) or,

ubasketball,f - utennis,f = - (4.47 - (- 4.47)) = -8.94 m/s ----------- (2)

Solving (1) and (2),

ubasketball,f = 2.87 m/s (i.e. upwards) and,

utennis,f = 11.81 m/s (i.e. upwards)

Now, let h2 be the height to which ball rebounds. Then, using equation of motion -

v2 - u2 = 2as

or, 0 - (11.81)2 = - 2 x 9.81 x h2

or, h2 = 7.1 m (height to which tennis ball rebounds)

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