2008M1. A skier of mass M is skiing down a frictionless hill that makes an angle θ with the horizontal, as shown in the diagram. The skier starts from rest at time t=0 and is subject to a velocity-dependent drag force due to air resistance of the form F=−bv, where v is the velocity of the skier and b is a positive constant. Express all algebraic answers in terms of M,b,θ, and fundamental constants.
a. On the dot below that represents the skier, draw a free-body diagram indicating and labeling all of the forces that act on the skier while the skier descends the hill.
b. Write a differential equation that can be used to solve for the velocity of the skier as a function of time.
c. Determine an expression for the terminal velocity vT of the skier.
d. Solve the differential equation in part (b) to determine the velocity of the skier as a function of time, showing all your steps.
e. On the axes below, sketch a graph of the acceleration a of the skier as a function of time t, and indicate the initial value of a. Take downhill as positive.
