A basketball player makes a jump shot. The 0.650-kg ball is released at a height of 1.90 m above the floor with a speed of 8.00 m/s. The ball goes through the net 3.10 m above the floor at a speed of 3.90 m/s. What is the work done on the ball by air resistance, a nonconservative force?
Initial KE+InitialPE-losses=finalKE+finalPE
To find the work done on the ball by air resistance, we need to calculate the change in the ball's kinetic energy. The work done by air resistance can be determined by subtracting the change in kinetic energy from the work done by gravity. Here's how you can calculate it step by step:
Step 1: Calculate the gravitational potential energy at the initial and final heights.
Since the ball is released from a height of 1.90 m above the floor, we can calculate the initial gravitational potential energy (PE_initial) using the formula:
PE_initial = m * g * h
Where:
m = mass of the ball = 0.650 kg
g = acceleration due to gravity = 9.8 m/s^2
h = initial height = 1.90 m
Step 2: Calculate the final gravitational potential energy (PE_final) at the height of the net.
PE_final = m * g * h_final
Where:
h_final = height of the net = 3.10 m
Step 3: Calculate the work done by gravity.
The work done by gravity is equal to the change in gravitational potential energy:
Work_gravity = PE_final - PE_initial
Step 4: Calculate the change in kinetic energy.
The change in kinetic energy (ΔKE) is the difference between the ball's initial and final kinetic energy.
Initial Kinetic Energy (KE_initial) = 0.5 * m * v_initial^2
Final Kinetic Energy (KE_final) = 0.5 * m * v_final^2
Where:
v_initial = initial speed = 8.00 m/s
v_final = final speed = 3.90 m/s
Step 5: Calculate the work done by air resistance.
The work done by air resistance is equal to the change in kinetic energy subtracted from the work done by gravity:
Work_air_resistance = Work_gravity - ΔKE
Now, plug in the numbers and calculate each step:
Step 1:
PE_initial = 0.650 kg * 9.8 m/s^2 * 1.90 m = 11.311 J
Step 2:
PE_final = 0.650 kg * 9.8 m/s^2 * 3.10 m = 19.026 J
Step 3:
Work_gravity = PE_final - PE_initial = 19.026 J - 11.311 J = 7.715 J
Step 4:
KE_initial = 0.5 * 0.650 kg * (8.00 m/s)^2 = 20.800 J
KE_final = 0.5 * 0.650 kg * (3.90 m/s)^2 = 5.333 J
ΔKE = KE_final - KE_initial = 5.333 J - 20.800 J = -15.467 J (Note: negative sign indicates a decrease in kinetic energy)
Step 5:
Work_air_resistance = Work_gravity - ΔKE = 7.715 J - (-15.467 J) = 23.182 J
Therefore, the work done on the ball by air resistance, a nonconservative force, is 23.182 Joules.