AP Free Response Question
2001 C2
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An explorer plans a mission to place a satellite into a circular orbit around the planet Jupiter, which has mass M
_{J}
= 1.90 x 10
^{27}
kg and radius R
_{J}
= 7.14 x 10
^{7}
m.
(a) If the radius of the planned orbit is
R
, use Newton's laws to show each of the following.
i. The orbital speed of the planned satellite is given by
ii. The period of the orbit is given by
(b) The explorer wants the satellite's orbit to be synchronized with Jupiter's rotation. This requires an equatorial orbit whose period equals Jupiter's rotation period of 9 hr 51 min = 3.55 x 10
^{4}
s. Determine the required orbital radius in meters.
(c) Suppose that the injection of the satellite into orbit is less than perfect. For an injection velocity that differs from the desired value in each of the following ways, sketch the resulting orbit on the figure. (
J
is the center of Jupiter, the dashed circle is the desired orbit, and
P
is the injection point.) Also, describe the resulting orbit qualitatively but specifically.
i. When the satellite is at the desired altitude over the equator, its velocity vector has the correct direction, but the speed is slightly
faster
than the correct speed for a circular orbit of that radius.
ii. When the satellite is at the desired altitude over the equator, its velocity vector has the correct direction, but the speed is slightly
slower
than the correct speed for a circular orbit of that radius.
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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