Using energy considerations, calculate the average force a 60.0 kg sprinter exerts backward on the track to accelerate from 2.00 to 6.00 m/s in a distance of 25.0 m, if he encounters a headwind that exerts an average force of 30.0 N against him.
To calculate the average force exerted by the sprinter, we need to consider the energy change involved in the acceleration. The work done on an object is equal to the change in its kinetic energy. The work done by the sprinter is equal to the force applied by the sprinter times the distance he travels.
Step 1: Calculate the initial kinetic energy (KE_initial) and final kinetic energy (KE_final).
- KE_initial = 0.5 * mass * velocity_initial^2
- KE_final = 0.5 * mass * velocity_final^2
Given:
- mass (m) = 60.0 kg
- velocity_initial (v_initial) = 2.00 m/s
- velocity_final (v_final) = 6.00 m/s
Substituting the values, we have:
- KE_initial = 0.5 * 60.0 kg * (2.00 m/s)^2
- KE_final = 0.5 * 60.0 kg * (6.00 m/s)^2
Step 2: Calculate the change in kinetic energy (ΔKE).
- ΔKE = KE_final - KE_initial
Substituting the values, we have:
- ΔKE = 0.5 * 60.0 kg * (6.00 m/s)^2 - 0.5 * 60.0 kg * (2.00 m/s)^2
Step 3: Calculate the work done by the sprinter (W_sprinter).
- W_sprinter = ΔKE
Step 4: Calculate the net work done on the sprinter (W_net).
- W_net = W_sprinter + W_wind
Given:
- W_wind = -30.0 N * 25.0 m (negative because it's against the sprinter)
Step 5: Calculate the average force exerted by the sprinter (F_sprinter).
- F_sprinter = W_net / distance
Substituting the values, we have:
- F_sprinter = (ΔKE - 30.0 N * 25.0 m) / 25.0 m
Now, calculate the values to get the final result.
W=KE2-KE1
=1/2*mv^2 - 1/2*mv^2
= 30kg(6m/s)^2 - 30kg(2m/s)^2
=1080 - 120
=960
W=(Fe - 30N)*d=960
(Fe - 30)25=960
25Fe - 750=960
25Fe=1710
Fe=68.4N