A basketball player makes a jump shot. The 0.600-kg ball is released at a height of 1.90 m above the floor with a speed of 7.40 m/s. The ball goes through the net 3.10 m above the floor at a speed of 4.40 m/s. What is the work done on the ball by air resistance, a nonconservative force?
To find the work done on the ball by air resistance, we need to calculate the change in mechanical energy of the ball during its trajectory. The work done by air resistance is equal to the difference in mechanical energy.
The mechanical energy of the ball consists of two components: the potential energy (PE) due to its height above the floor and the kinetic energy (KE) due to its motion.
1. Calculate the potential energy (PE1) of the ball at the initial height of 1.90 m:
PE1 = m * g * h
where m is the mass of the ball (0.600 kg), g is the gravitational acceleration (9.8 m/s²), and h is the initial height (1.90 m).
PE1 = 0.600 kg * 9.8 m/s² * 1.90 m
2. Calculate the kinetic energy (KE1) of the ball at the initial speed of 7.40 m/s:
KE1 = (1/2) * m * v^2
where m is the mass of the ball (0.600 kg) and v is the initial speed (7.40 m/s).
KE1 = (1/2) * 0.600 kg * (7.40 m/s)^2
3. Calculate the potential energy (PE2) of the ball at the final height of 3.10 m (height of the net):
PE2 = m * g * h
where m is the mass of the ball (0.600 kg), g is the gravitational acceleration (9.8 m/s²), and h is the final height (3.10 m).
PE2 = 0.600 kg * 9.8 m/s² * 3.10 m
4. Calculate the kinetic energy (KE2) of the ball at the final speed of 4.40 m/s:
KE2 = (1/2) * m * v^2
where m is the mass of the ball (0.600 kg) and v is the final speed (4.40 m/s).
KE2 = (1/2) * 0.600 kg * (4.40 m/s)^2
5. Calculate the change in mechanical energy (ΔE) of the ball:
ΔE = (PE2 + KE2) - (PE1 + KE1)
ΔE = (0.600 kg * 9.8 m/s² * 3.10 m) + ((1/2) * 0.600 kg * (4.40 m/s)^2) - (0.600 kg * 9.8 m/s² * 1.90 m) - ((1/2) * 0.600 kg * (7.40 m/s)^2)
The work done on the ball by air resistance is equal to the change in mechanical energy of the ball, ΔE.