PROBLEM 13.111* (Continued)
PROBLEM 13.112
Show that the values vA and vP of the speed of an earth satellite at the
apogee A and the perigee P of an elliptic orbit are defined by the
relations
22
22
PA
AP
APA APP
rr
GM GM
vv
r rr r rr
= =
++
where M is the mass of the earth, and
A
r
and
P
r
represent,
respectively, the maximum and minimum distances of the orbit to the
center of the earth.
SOLUTION
A
A PA
+
PROBLEM 13.113
Show that the total energy E of an earth satellite of mass m
describing an elliptic orbit is
/( ),
AP
E GMm r r=−+
where M is the
mass of the earth, and
A
r
and
P
r
represent, respectively, the
maximum and minimum distances of the orbit to the center of the
earth. (Recall that the gravitational potential energy of a satellite was
defined as being zero at an infinite distance from the earth.)
PROBLEM 13.114*
A space probe describes a circular orbit of radius nR with a velocity v0 about a planet of radius R and center O.
Show that (a) in order for the probe to leave its orbit and hit the planet at an angle
θ
with the vertical, its
velocity must be reduced to
α
v0, where
22
2( 1)
sin sin
n
n
αθ θ
−
=−
(b) the probe will not hit the planet if
α
is larger than
2 /(1 ) .n+
0
0
sin
nv
v
α
θ
=
(2)
Replacing v in (1) by (2)
2
20
0
22
nv
GM GM
α
PROBLEM 13.114* (Continued)
PROBLEM 13.115
A missile is fired from the ground with an initial velocity
0
v
forming an angle
0
φ
with the vertical. If the
missile is to reach a maximum altitude equal to
,R
α
where R is the radius of the earth, (a) show that the
required angle
0
φ
is defined by the relation
2
esc
0
0
sin (1 ) 1 1
v
v
α
φα α
=+−
+
where
esc
v
is the escape velocity, (b) determine the range of allowable values of
0
.v
SOLUTION
0
0
v
+
PROBLEM 13.115 (Continued)
PROBLEM 13.116
A spacecraft of mass m describes a circular orbit of radius
1
r
around
the earth. (a) Show that the additional energy
E∆
which must be
imparted to the spacecraft to transfer it to a circular orbit of larger
radius
2
r
is
21
12
()
2
GMm r r
Err
−
∆=
where M is the mass of the earth. (b) Further show that if the transfer
from one circular orbit to the other is executed by placing the
spacecraft on a transitional semielliptic path AB, the amounts of
energy
A
E∆
and
B
E∆
which must be imparted at A and B are,
respectively, proportional to
2
r
and
1
:r
2
12
A
r
EE
rr
∆= ∆
+
1
12
B
r
EE
rr
∆= ∆
+
E∆
1
1 21
()
PROBLEM 13.116 (Continued)
PROBLEM 13.117*
Using the answers obtained in Problem 13.108, show that the intended circular orbit and the resulting elliptic
orbit intersect at the ends of the minor axis of the elliptic orbit.
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.
2
0
sin
cos
Cr
α
α
=
PROBLEM 13.117* (Continued)
GM
PROBLEM 13.118*
(a) Express in terms of
min
r
and
max
v
the angular momentum per unit mass, h, and the total energy per unit
mass, E/m, of a space vehicle moving under the gravitational attraction of a planet of mass M (Figure 13.15).
(b) Eliminating
max
v
between the equations obtained, derive the formula
2
2
min
12
11
GM E h
r m GM
h
= ++
(c) Show that the eccentricity
ε
of the trajectory of the vehicle can be expressed as
2
2
1Eh
m GM
ε
= +
(d) Further show that the trajectory of the vehicle is a hyperbola, an ellipse, or a parabola, depending on
whether E is positive, negative, or zero.
22
min min
rr
hh
PROBLEM 13.118* (Continued)
PROBLEM 13.119
A 35,000 Mg ocean liner has an initial velocity of 4 km/h. Neglecting the frictional resistance of the water,
determine the time required to bring the liner to rest by using a single tugboat which exerts a constant force of
150 kN.
SOLUTION
6
PROBLEM 13.120
A 2500-lb automobile is moving at a speed of 60 mi/h when the brakes are fully applied, causing all four
wheels to skid. Determine the time required to stop the automobile (a) on dry pavement (
m
k = 0.75), (b) on an
icy road (
m
k = 0.10).
PROBLEM 13.121
A sailboat weighing 980 lb with its occupants is running down
wind at 8 mi/h when its spinnaker is raised to increase its
speed. Determine the net force provided by the spinnaker over
the 10-s interval that it takes for the boat to reach a speed of
12 mi/h.
PROBLEM 13.122
A truck is hauling a 300-kg log out of a ditch
using a winch attached to the back of the truck.
Knowing the winch applies a constant force of
2500 N and the coefficient of kinetic friction
between the ground and the log is 0.45,
determine the time for the log to reach a speed
of 0.5m/s.
SOLUTION
Apply the principle of impulse and momentum to the log.
PROBLEM 13.123
The coefficients of friction between the load and the flatbed
trailer shown are
0.40
s
m
=
and
0.35.
k
m
=
Knowing that the
speed of the rig is
55 mi/h,
determine the shortest time in
which the rig can be brought to a stop if the load is not to shift.
SOLUTION
PROBLEM 13.124
Steep safety ramps are built beside mountain highways to
enable vehicles with defective brakes to stop. A 10-ton truck
enters a 15° ramp at a high speed v0 = 108 ft/s and travels for 6
s before its speed is reduced to 36 ft/s. Assuming constant
deceleration, determine (a) the magnitude of the braking force,
(b) the additional time required for the truck to stop. Neglect air
resistance and rolling resistance.
SOLUTION
20,000 lbW=
2
20,000 621.118 lb s /ft
32.2
m= = ⋅
Momentum in the x direction
01
: ( sin15 )x mv F mg t mv− + °=
621.118(108) ( sin15 )6 (621.118)(36)F mg− + °=
(b)
0
( sin15 ) 0mv F mg t− + °=
t = total time
621.118(108) 7453.4 0;t−=
t = 9.00 s
Additional time = 9 – 6
3.00 st=
PROBLEM 13.125
Baggage on the floor of the baggage car of a high-speed train is not prevented from moving other than by
friction. The train is travelling down a 5 percent grade when it decreases its speed at a constant rate from
120 mi/h to 60 mi/h in a time interval of 12 s. Determine the smallest allowable value of the coefficient of
static friction between a trunk and the floor of the baggage car if the trunk is not to slide.
SOLUTION