Answer :
Final answer:
To find the speed of the crate at 15 m, the work done by the motor force on the cable is calculated and equated to the change in kinetic energy of the crate, resulting in a final speed of approximately 14.74 m/s.
Explanation:
The question asks us to determine the speed of a 145-kg crate when it rises to s = 15 m, given that the force exerted by the motor on the cable is F = (600 + 2s2) N. Since the crate starts from rest, we can use work-energy principle to solve this problem.
The work done by the force is equal to the change in kinetic energy of the crate. First, we calculate the work done (W) as the integral of the force over the displacement s:
W = ∫ F ds = ∫ (600 + 2s2) ds from 0 to 15 m = [600s + (2/3)s3] from 0 to 15 = 600(15) + (2/3)(15)3 = 9000 + 2(3375) = 9000 + 6750 = 15750 J
The initial kinetic energy (KEi) is 0 since the crate starts from rest, therefore the final kinetic energy (KEf) is equal to the work done, KEf = W. Now we can find the final speed (v) of the crate using the kinetic energy formula KE = 0.5 m v2:
15750 J = 0.5 * 145 kg * v2
v = √(2 * 15750 J / 145 kg)
v = √(217.24)
v ≈ 14.74 m/s
Thus, the speed of the crate when it rises to 15 m is approximately 14.74 m/s.
