PROBLEM 13.99 (Continued)
1.661
m
min 5.21 m/sv=
PROBLEM 13.100
A spacecraft is describing an elliptic orbit of minimum altitude
2400
A
h=
km and maximum altitude
9600
B
h=
km above
the surface of the earth. Determine the speed of the spacecraft
at A.
SOLUTION
6370 km 2400 km
8770 km
A
A
r
r
= +
=
PROBLEM 13.101
While describing a circular orbit, 185 mi above the surface of
the earth, a space shuttle ejects at Point A an inertial upper stage
(IUS) carrying a communication satellite to be placed in a
geosynchronous orbit (see Problem 13.87) at an altitude of
22,230 mi above the surface of the earth. Determine (a) the
velocity of the IUS relative to the shuttle after its engine has
been fired at A, (b) the increase in velocity required at B to
place the satellite in its final orbit.
SOLUTION
6
PROBLEM 13.101 (Continued)
A
B
B
PROBLEM 13.102
A spacecraft approaching the planet Saturn reaches Point A with a
velocity
A
v
of magnitude
3
68.8 10 ft/s.×
It is to be placed in an
elliptic orbit about Saturn so that it will be able to periodically
examine Tethys, one of Saturn’s moons. Tethys is in a circular
orbit of radius
3
183 10 mi×
about the center of Saturn, traveling at
a speed of
3
37.2 10 ft/s.×
Determine (a) the decrease in speed
required by the spacecraft at A to achieve the desired orbit, (b) the
speed of the spacecraft when it reaches the orbit of Tethys at B.
SOLUTION
PROBLEM 13.102 (Continued)
PROBLEM 13.103
A spacecraft traveling along a parabolic path toward the planet Jupiter is
expected to reach Point A with a velocity vA of magnitude 26.9 km/s. Its
engines will then be fired to slow it down, placing it into an elliptic orbit
which will bring it to within 100 × 103 km of Jupiter. Determine the
decrease in speed
v∆
at Point A which will place the spacecraft into the
required orbit. The mass of Jupiter is 319 times the mass of the earth.
SOLUTION
PROBLEM 13.103 (Continued)
PROBLEM 13.104
As a first approximation to the analysis of a space flight from the
earth to Mars, it is assumed that the orbits of the earth and Mars are
circular and coplanar. The mean distances from the sun to the
earth and to Mars are
6
149.6 10×
km and
6
227.8 10×
km,
respectively. To place the spacecraft into an elliptical transfer orbit
at Point A, its speed is increased over a short interval of time to
A
v
which is faster than the earth’s orbital speed. When the spacecraft
reaches Point B on the elliptical transfer orbit, its speed
B
v
is
increased to the orbital speed of Mars. Knowing that the mass of
the sun is
3
332.8 10×
times the mass of the earth, determine the
increase in velocity required (a) at A, (b) at B.
SOLUTION
mass of the sunM=
3 2 6 2 20 3 2
332.8(10) (9.81 m/s )(6.37 10 m) 1.3247(10) m /sGM = ×=
Circular orbits
9
Earth 29.758 m/s
149.6(10)
E
GM
v= =
9
Mars 24.115 m/s
227.8(10)
M
GM
v= =
Conservation of angular momentum
Elliptical orbit
(149.6) (227.8)
AB
vv=
Conservation of energy
22
99
11
22
149.6(10) 227.8(10)
AB
GM GM
vv−=−
(227.8) 1.52273
(149.6)
AB B
vv v= =
20 20
22 2
99
1 1.3247(10) 1 1.3247(10)
(1.52273)
22
149.6(10) 227.8(10)
BB
vv−=−
28
0.65935 3.0398(10)
B
v=
28
4.6102(10)
B
v=
21, 471 m/s, 32,695 m/s
BA
vv= =
(a) Increase at A,
32.695 29.758 2.94 km/s
AE
vv−= − =
(b) Increase at B,
24.115 21.471 2.64 km/s
BM
vv−= − =
PROBLEM 13.105
The optimal way of transferring a space vehicle from an inner circular
orbit to an outer coplanar circular orbit is to fire its engines as it passes
through A to increase its speed and place it in an elliptic transfer orbit.
Another increase in speed as it passes through B will place it in the
desired circular orbit. For a vehicle in a circular orbit about the earth at
an altitude
1200h=
mi, which is to be transferred to a circular orbit at
an altitude
2
500h=
mi, determine (a) the required increases in speed
at A and at B, (b) the total energy per unit mass required to execute the
transfer.
SOLUTION
B
PROBLEM 13.105 (Continued)
PROBLEM 13.106
During a flyby of the earth, the velocity of a spacecraft is 10.4 km/s
as it reaches its minimum altitude of 990 km above the surface at
Point A. At Point B the spacecraft is observed to have an altitude of
8350 km. Determine (a) the magnitude of the velocity at Point B,
(b) the angle
.
B
φ
SOLUTION
46 6
PROBLEM 13.107
A space platform is in a circular orbit about the earth at an
altitude of 300 km. As the platform passes through A, a rocket
carrying a communications satellite is launched from the
platform with a relative velocity of magnitude 3.44 km/s in a
direction tangent to the orbit of the platform. This was intended
to place the rocket in an elliptic transfer orbit bringing it to
Point B, where the rocket would again be fired to place the
satellite in a geosynchronous orbit of radius 42,140 km. After
launching, it was discovered that the relative velocity imparted
to the rocket was too large. Determine the angle
γ
at which the
rocket will cross the intended orbit at Point C.
SOLUTION
6
6
6370 km
6370 km 300 km
6.67 10 m
A
A
R
r
r
=
= +
= ×
PROBLEM 13.107 (Continued)
PROBLEM 13.108
A satellite is projected into space with a velocity v0 at a distance r0 from the
center of the earth by the last stage of its launching rocket. The velocity v0
was designed to send the satellite into a circular orbit of radius r0.
However, owing to a malfunction of control, the satellite is not projected
horizontally but at an angle
α
with the horizontal and, as a result, is
propelled into an elliptic orbit. Determine the maximum and minimum
values of the distance from the center of the earth to the satellite.
AA
rr
PROBLEM 13.108 (Continued)
max 0
min 0
PROBLEM 13.109
A space vehicle is to rendezvous with an orbiting laboratory which circles
the earth at a constant altitude of 360 km. The shuttle has reached an
altitude of 60 km when its engine is shut off, and its velocity v0 forms an
angle
φ
0 = 50° with the vertical OB at that time. What magnitude should v0
have if the shuttle’s trajectory is to be tangent at A to the orbit of the
laboratory?
SOLUTION
0
0
PROBLEM 13.110
A space vehicle is in a circular orbit at an altitude of 225 mi above the
earth. To return to earth, it decreases its speed as it passes through A by
firing its engine for a short interval of time in a direction opposite to the
direction of its motion. Knowing that the velocity of the space vehicle
should form an angle
60°
B
φ
=
with the vertical as it reaches Point B at an
altitude of 40 mi, determine (a) the required speed of the vehicle as it
leaves its circular orbit at A, (b) its speed at Point B.
PROBLEM 13.110 (Continued)
Conservation of angular momentum:
sin
AA BB B
r mv r mv
φ
=
PROBLEM 13.111*
In Problem 13.110, the speed of the space vehicle was decreased as it passed through A by firing its engine in
a direction opposite to the direction of motion. An alternative strategy for taking the space vehicle out of its
circular orbit would be to turn it around so that its engine would point away from the earth and then give it an
incremental velocity
A
∆v
toward the center O of the earth. This would likely require a smaller expenditure of
energy when firing the engine at A, but might result in too fast a descent at B. Assuming this strategy is used
with only 50 percent of the energy expenditure used in Problem 13.109, determine the resulting values of
B
φ
and
.
B
v
SOLUTION
6
3960 mi 225 mi
4185 mi 22.097 10 ft
3960 mi 40 mi 4000 mi
A
A
B
r
r
r
= +
= = ×
= +=
32 32
109
6
109
6
110 109
22
11
(25.24 10 ) (11.32 10 )
22
254.46 10 ft lb
(254.46 10 )
(0.50) ft lb
2
Em m
Em
m
EE
∆= × − ×
∆= × ⋅
×
∆= ∆= ⋅