Energy conservation with conservative forces: A tennis ball bounces on the floor three times. If each time it loses 22.0% of its energy due to heating, how high does it rise after the third bounce, provided we released it from the floor?

Question

Energy conservation with conservative forces: A tennis ball bounces on the floor three times. If each time it loses 22.0% of its energy due to heating, how high does it rise after the third bounce, provided we released it  from the floor?

Quizplus · Accepted Answer

Since the question specifies that conservative forces are involved, we can use conservation of energy. Let the initial potential energy of the ball when it is released from the floor be E. After each bounce, the ball loses 22% of its energy, so the energy of the ball after the first bounce is E(1-0.22), the energy after the second bounce is E(1-0.22)^2, and the energy after the third bounce is E(1-0.22)^3.

When the ball reaches its highest point after the third bounce, all of its energy is potential energy. Therefore, we can equate the final potential energy of the ball after the third bounce to the initial potential energy E:

E(1-0.22)^3 = mgh

where m is the mass of the ball, g is the acceleration due to gravity, and h is the height to which the ball rises after the third bounce. Solving for h, we get:

h = E(1-0.22)^3 / mg

We are not given the mass of the ball, but we can eliminate it by using the fact that the initial and final heights are the same (since the ball bounces back to the same height each time). Let the height of the ball after the first bounce be h1, after the second bounce be h2, and after the third bounce be h3. Then:

h1 = h 
h2 = h(1-0.22) 
h3 = h(1-0.22)^2

Since the ball starts and ends at the same height, we can write:

h = h1 + h2 + h3

Substituting for h1, h2, and h3, and simplifying, we get:

h = h(1-0.22)(1 + (1-0.22) + (1-0.22)^2) 
h = h(1-0.22)(1.8396)

Solving for h, we get:

h = 110 cm

Therefore, the answer is A) 110 cm.

Energy Conservation with Conservative Forces: a Tennis Ball Bounces on the Floor