A box of mass 200 kg is moving in the x-direction with a constant speed of 40 m/s. Aretarding force F = − 20x J/m moves it from x = 10 m to x = 20 m. Estimate its finalkinetic energy

To solve this problem, we need to use the work-energy theorem which states that the work done on an object is equal to the change in its kinetic energy.

Step 1: Calculate the initial kinetic energy. The formula for kinetic energy is KE = 1/2 * m * v^2.

Given: m = 200 kg (mass) v = 40 m/s (velocity)

Substitute the given values into the formula:

KE_initial = 1/2 * 200 kg * (40 m/s)^2 = 160,000 J (Joules)

Step 2: Calculate the work done by the retarding force. The formula for work done is W = F * d, where F is the force and d is the distance.

Given: F = -20x J/m (force) d = 20 m - 10 m = 10 m (distance)

The force is not constant, but varies with x. So, the work done by the force is the integral of F dx from x = 10 m to x = 20 m.

W = ∫ from 10 to 20 of -20x dx = [-10x^2] from 10 to 20 = -10*(20^2) - [-10*(10^2)] = -3000 J

Step 3: Calculate the final kinetic energy. According to the work-energy theorem, the work done on the box is equal to the change in its kinetic energy. So, the final kinetic energy is the initial kinetic energy plus the work done on the box.

KE_final = KE_initial + W = 160,000 J - 3000 J = 157,000 J

So, the final kinetic energy of the box is 157,000 Joules.