AP Physics C:
Gravitation
Problems
1984M2. Two satellites, of masses m and 3m, respectively, are in the same circular orbit about the Earth's center, as
shown in the diagram above. The Earth has mass Me and radius Re. In this orbit, which has a radius of 2Re, the
satellites initially move with the same orbital speed vo but in opposite directions.
a. Calculate the orbital speed vo of the satellites in terms of G, Me, and Re.
b. Assume that the satellites collide head-on and stick together. In terms of vo find the speed v of the
combination immediately after the collision.
c.
Calculate the total mechanical energy of the system immediately after the collision in terms of G, m, Me, and
Re. Assume that the gravitational potential energy of an object is defined to be zero at an infinite distance from
the Earth.
1992M3. A spacecraft of mass 1,000 kilograms is in an elliptical orbit about the Earth, as shown above. At point
A the spacecraft is at a distance rA = 1.2 x 107 meters from the center of the Earth and its velocity, of magnitude VA
= 7.1 x 103 meters per second, is perpendicular to the line connecting the center of the Earth to the spacecraft. The
mass and radius of the Earth are ME = 6.0 X 1024 kilograms and rE = 6.4 X 106 meters, respectively.
Determine each of the following for the spacecraft when it is at point A .
a. The total mechanical energy of the spacecraft, assuming that the gravitational potential energy is zero at an
infinite distance from the Earth.
b.
The magnitude of the angular momentum of the spacecraft about the center of the Earth.
Later the spacecraft is at point B on the exact opposite side of the orbit at a distance r B = 3.6 X 107
meters from the center of the Earth.
c. Determine the speed vB of the spacecraft at point B.
Suppose that a different spacecraft is at point A, a distance r A = 1.2 X 107 meters from the center of the Earth.
Determine each of the following.
d. The speed of the spacecraft if it is in a circular orbit around the Earth
e.
The minimum speed of the spacecraft at point A if it is to escape completely from the Earth
1994M3. A satellite of mass m is in an elliptical orbit around the Earth, which has mass M e and radius Re. The
orbit varies from closest approach of a at point A to maximum distance of b from the center of the Earth at
point B. At point A, the speed of the satellite is vo Assume that the gravitational potential energy Ug = 0 when
masses are an infinite distance apart. Express your answers in terms of a, b, m, Me, Re, vo, and G.
a. Write the appropriate definite integral, including limits, that can be evaluated to show that the potential
energy of the satellite when it is a distance r from the center of the Earth is given by
Ug = -Gmem
r
b. Determine the total energy of the satellite when it is at A.
c. What is the magnitude of the angular momentum of the satellite about the center of the Earth when it is at
A?
d. Determine the velocity of the satellite as it passes point B in its orbit.
As the satellite passes point A, a rocket engine on the satellite is fired so that its orbit is changed to a circular
orbit of radius a about the center of the Earth.
e. Determine the speed of the satellite for this circular orbit.
f. Determine the work done by the rocket engine to effect this change.
1995M3. Two stars, A and B. are in circular orbits of radii r a and rb, respectively, about their common
center of mass at point P, as shown above. Each star has the same period of revolution T.
Determine expressions for the following three quantities in terms of r a, rb, T, and fundamental
constants.
a. The centripetal acceleration of star A
b. The mass Mb of star B
c. The mass Ma of star A
Determine expressions for the following two quantities in terms of Ma, Mb, ra, rb, T, and fundamental
constants.
d. The moment of inertia of the two-star system about its center of mass.
e. The angular momentum of the system about the center of mass.
1999M2 A spherical, nonrotating planet has a radius R and a uniform density  throughout its volume. Suppose a
narrow tunnel were drilled through the planet along one of its diameters, as shown in the figure above, in which a
small ball of mass m could move freely under the influence of gravity. Let r be the distance of the ball from the
center of the planet.
a. Show that the magnitude of the force on the ball at a distance r < R from the center of the planet is given
by F = -Cr. where C = 4/3(Gm).
b.
On the axes below, sketch the force F on the ball as a function of distance r from the center of the planet.
The ball is dropped into the tunnel from rest at point P at the planet's surface.
c. Determine the work done by gravity as the ball moves from the surface to the center of the planet.
d.
Determine the speed of the ball when it reaches the center of the planet.
e.
Fully describe the subsequent motion of the ball from the time it reaches the center of the planet.
f.
Write an equation that could be used to calculate the time it takes the ball to move from point P to the center of
the planet. It is not necessary to solve this equation.
2001M2. An explorer plans a mission to place a satellite into a circular orbit around the planet Jupiter, which
has mass MJ = 1.90 x 1027 kg and radius RJ = 7.14 x 107 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
v
ii.
GM J
R
The period of the orbit is given by
4 2 R 3
T
GM J
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.