A tennis ball (m � 57 g) is projected vertically with an

initial speed of 8.8 m/s. (a) If the ball rises to a maximum height
of 3.7 m, how much kinetic energy was dissipated by the drag
force of air resistance? (b) How much higher would the ball
have gone in a vacuum?

518.94 J

To find the amount of kinetic energy dissipated by the drag force of air resistance, we need to know the work done by the drag force. The work done can be determined by calculating the change in kinetic energy of the tennis ball.

(a) To find the change in kinetic energy, we need to calculate the initial kinetic energy (KEi) and the final kinetic energy (KEf).

The initial kinetic energy can be calculated using the formula:

KEi = 0.5 * mass * velocity^2

Substituting the given values:

KEi = 0.5 * 0.057 kg * (8.8 m/s)^2
= 0.5 * 0.057 kg * 77.44 m^2/s^2
= 2.2128 J

Next, let's calculate the final kinetic energy. At the maximum height, the ball's velocity becomes zero, so the final kinetic energy (KEf) is zero.

Therefore, the change in kinetic energy (ΔKE) is:

ΔKE = KEf - KEi
= 0 - 2.2128 J
= -2.2128 J

Since the value of ΔKE is negative, we deduce that the kinetic energy is dissipated by the drag force of air resistance. So the answer to part (a) is -2.2128 J.

(b) To determine how much higher the ball would have gone in a vacuum, we need to calculate the potential energy at the maximum height in both scenarios - with air resistance and in a vacuum.

The potential energy (PE) at the maximum height can be calculated using the formula:

PE = mass * gravitational acceleration * height

Substituting the given values (mass = 0.057 kg, gravitational acceleration = 9.8 m/s^2, height = 3.7 m):

PE = 0.057 kg * 9.8 m/s^2 * 3.7 m
= 2.12066 J

Therefore, in a vacuum, the potential energy at the maximum height would be 2.12066 J.

To find the additional height in a vacuum, we can use the formula:

Additional height = PE / mass / gravitational acceleration

Substituting the values:

Additional height = 2.12066 J / 0.057 kg / 9.8 m/s^2
= 3.704 m

Therefore, the ball would have gone an additional height of 3.704 m in a vacuum.