A skier is pulled by a towrope up a frictionless ski slope that makes an angle of 12° with the horizontal. The rope moves parallel to the slope with a constant speed of 1.0 m/s. The force of the rope does 900 J of work on the skier as the skier moves a distance of 8.0 m up the incline. (a) If the rope moved with a constant speed of 2.0 m/s, how much work would the force of the rope do on the skier as the skier moved a distance of 8.0 m up the incline? At what rate is the force of the rope doing work on the skier when the rope moves with a  speed of (b) 1.0 m/s and (c) 2.0 m/s?

The work done by a force is given by the equation W = Fd, where W is the work done, F is the force, and d is the distance.

(a) In the first scenario, we know that the work done is 900 J and the distance is 8.0 m. We can rearrange the equation to solve for the force: F = W/d = 900 J / 8.0 m = 112.5 N.

If the speed of the rope changes, this does not affect the force exerted by the rope or the distance the skier travels, so the work done would remain the same. Therefore, if the rope moved with a constant speed of 2.0 m/s, the work done on the skier as the skier moved a distance of 8.0 m up the incline would still be 900 J.

(b) The rate at which work is done, or power, is given by the equation P = W/t, where P is power, W is work, and t is time. In this case, we need to find the time it takes for the skier to travel 8.0 m at a speed of 1.0 m/s. The time is given by t = d/v = 8.0 m / 1.0 m/s = 8.0 s. Substituting these values into the power equation gives P = 900 J / 8.0 s = 112.5 W.

(c) If the speed of the rope increases to 2.0 m/s, the time it takes for the skier to travel 8.0 m decreases to t = d/v = 8.0 m / 2.0 m/s = 4.0 s. Substituting these values into the power equation gives P = 900 J / 4.0 s = 225 W. Therefore, the rate at which the force of the rope is doing work on the skier when the rope moves with a speed of 2.0 m/s is 225 W.