AP Free Response Question
1999 C1
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In a laboratory experiment, you wish to determine the initial speed of a dart just after it leaves a dart gun. The dart, of mass
m
, is fired with the gun very close to a wooden block of mass
M
_{o}
which hangs from a cord of length
and negligible mass, as shown above. Assume the size of the block is negligible compared to
, and the dart is moving horizontally when it hits the left side of the block at its center and becomes embedded in it. The block swings up to a maximum angle
_{ }
from the vertical. Express your answers to the following in terms of
m
,
M
_{o}
,
,
and
g
.
Determine the speed v
_{0}
of the dart immediately before it strikes the block.
The dart and block subsequently swing as a pendulum. Determine the tension in the cord when it returns to the lowest point of the swing.
At your lab table you have only the following additional equipment.
Without destroying or disassembling any of this equipment, design another practical method for determining the speed of the dart just after it leaves the gun. Indicate the measurements you would take, and how the speed could be determined from these measurements.
The dart is now shot into a block of wood that is fixed in place. The block exerts a force
F
on the dart that is proportional to the dart’s velocity
v
and in the opposite direction, that is
F = bv
, where
b
is a constant.
Derive an expression for the distance
L
that the dart penetrates into the block in terms of
m
,
v
_{0}
, and
b
.
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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