AP Free Response Question 2010 C3

A skier of mass m will be pulled up a hill by a rope, as shown above. The magnitude of the acceleration of the skier as a function of time t can be modeled by the equations

where amax and T are constants. The hill is inclined at an angle q above the horizontal, and friction between the skis and the snow is negligible. Express your answers in terms of given quantities and fundamental constants.

 (a) Derive an expression for the velocity of the skier as a function of time during the acceleration. Assume the skier starts from rest.

 (b) Derive an expression for the work done by the net force on the skier from rest until terminal speed is reached.

 (c) Determine the magnitude of the force exerted by the rope on the skier at terminal speed.

 (d) Derive an expression for the total impulse imparted to the skier during the acceleration.

 (e) Suppose that the magnitude of the acceleration is instead modeled as for all t > 0 , where amax and T are the same as in the original model.   On the axes below, sketch the graphs of the force exerted by the rope on the skier for the two models, from t = 0 to a time t > T. Label the original model F1 and the new model F2 .