AP Free Response Question
2010 C3
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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
a
_{max}
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
a
_{max}
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 F
_{1}
and the new model F
_{2}
.
Topic Formulas
Description
Published Formula
angular displacement
angular momentum
angular velocity
center of mass
centripetal acceleration
elastic potential energy
friction
gravitational force (vector)
gravitational potential energy
gravitational potential energy
Hooke's Law
impulse
impulse
kinetic energy
linear momentum
linear velocity and angular velocity
moment of inertia
net torque
Newton's 2nd Law
Newton's Law of Universal Gravitation
period and frequency
period of a simple pendulum
period of a spring
potential energy
power
power
power (dot product)
rate of change of momentum
rate of change of work
rotational kinetic energy
torque
uniform acceleration - displacement and instantaneous velocity
uniform acceleration - instantaneous position
uniform acceleration - instantaneous velocity
work
work (dot product)
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