AP Free Response Question
2000 C2
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A rubber ball of mass
m
is dropped from a cliff. As the ball falls, it is subject to air drag (a resistive force caused by the air). The drag force on the ball has magnitude
bv
^{2}
, where
b
is a constant drag coefficient and
v
is the instantaneous speed of the ball. The drag coefficient
b
is directly proportional to the cross sectional area of the ball and the density of the air and does not depend on the mass of the ball. As the ball falls, its speed approaches a constant value called the terminal speed.
(a) On the figure below, draw and label all the forces on the ball at some instant before it reaches terminal speed
(b) State whether the magnitude of the acceleration of the ball of mass m increases, decreases, or remains the same as the ball approaches terminal speed. Explain.
(c) Write, but do NOT solve, a differential equation for the instantaneous speed
v
of the ball in terms of time
t
, the given quantities, and fundamental constants.
(d) Determine the terminal speed
v
_{t}
in terms of the given quantities and fundamental constants.
(e) Determine the energy dissipated by the drag force during the fall if the ball is released at height
h
and reaches its terminal speed before hitting the ground, in terms of the given quantities and fundamental constants.
Topic Formulas
Description
Published Formula
angular displacement
angular momentum
angular velocity
center of mass
centripetal acceleration
friction
gravitational force (vector)
gravitational potential energy
Hooke's Law
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 elastic energy
potential energy
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 (dot product)
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