368
CHAPTE R 13 GRAVITATION
13-6 PLANETS AND SATELLITES: KEPLER’S LAWS
After reading this module, you should be able to . . .
13.17 Identify Kepler’s three laws.
13.18 Identify which of Kepler’s laws is equivalent to the law
of conservation of angular momentum.
13.19 On a sketch of an elliptical orbit, identify the semimajor
axis, the eccentricity, the perihelion, the aphelion, and the
focal points.
13.20 For an elliptical orbit, apply the relationships between
the semimajor axis, the eccentricity, the perihelion, and the
aphelion.
13.21 For an orbiting natural or artificial satellite, apply
Kepler’s relationship between the orbital period and radius
and the mass of the astronomical body being orbited.
● The motion of satellites, both natural and artificial, is governed by Kepler’s laws:
1. The law of orbits. All planets move in elliptical orbits with
the Sun at one focus.
2. The law of areas. A line joining any planet to the Sun
sweeps out equal areas in equal time intervals. (This
statement is equivalent to conservation of angular
momentum.)
3. The law of periods. The square of the period T of any
planet is proportional to the cube of the semimajor axis a
of its orbit. For circular orbits with radius r,
(law of periods),
where M is the mass of the attracting body—the Sun in the
case of the solar system. For elliptical planetary orbits, the
semimajor axis a is substituted for r.
T
2 ϭ
4
2
GM r
3
Learning Objectives
Key Ideas
tially at distance 10R E and finally at distance R E , where R E is
Earth’s radius (6.37 ϫ 10
6
m). Substituting Eq. 13-21 for U
and
for K, we rewrite Eq. 13-29 as
Rearranging and substituting known values, we find
ϭ (12 ϫ 10
3
m/s)
2
ϭ 2.567 ϫ 10
8
m
2
/s
2
,
and
v f ϭ 1.60 ϫ 10
4
m/s ϭ 16 km/s.
(Answer)
At this speed, the asteroid would not have to be particularly large to do considerable damage at impact. If it
were only 5 m across, the impact could release about as
much energy as the nuclear explosion at Hiroshima.
Alarmingly, about 500 million asteroids of this size are
near Earth’s orbit, and in 1994 one of them apparently
penetrated Earth’s atmosphere and exploded 20 km
above the South Pacific (setting off nuclear-explosion
warnings on six military satellites).
ϩ
2(6.67 ϫ 10
Ϫ11
m
3
/kgиs
2
)(5.98 ϫ 10
24
kg)
6.37 ϫ 10
6
m
0.9
v
2
f ϭ v
2
i ϩ
2GM
R E
1 Ϫ
1
10
1
2 mv
2
f Ϫ
GMm
R E
ϭ
1
2 mv
2
i Ϫ
GMm
10R E
.
1
2 mv
2
Sample Problem 13.03 Asteroid falling from space, mechanical energy
An asteroid, headed directly toward Earth, has a speed of
12 km/s relative to the planet when the asteroid is 10 Earth
radii from Earth’s center. Neglecting the effects of Earth’s
atmosphere on the asteroid, find the asteroid’s speed v f
when it reaches Earth’s surface.
KEY IDEAS
Because we are to neglect the effects of the atmosphere on
the asteroid, the mechanical energy of the asteroid – Earth
system is conserved during the fall. Thus, the final mechanical energy (when the asteroid reaches Earth’s surface) is
equal to the initial mechanical energy. With kinetic energy K
and gravitational potential energy U, we can write this as
K f ϩ U f ϭ K i ϩ U i .
( 1 3 - 2 9 )
Also, if we assume the system is isolated, the system’s linear momentum must be conserved during the fall. Therefore,
the momentum change of the asteroid and that of Earth must
be equal in magnitude and opposite in sign. However, because
Earth’s mass is so much greater than the asteroid’s mass, the
change in Earth’s speed is negligible relative to the change in
the asteroid’s speed. So, the change in Earth’s kinetic energy is
also negligible.Thus, we can assume that the kinetic energies in
Eq. 13-29 are those of the asteroid alone.
Calculations: Let m represent the asteroid’s mass and M
represent Earth’s mass (5.98 ϫ 10
24
kg). The asteroid is iniAdditional examples, video, and practice available at WileyPLUS
CHAPTE R 13 GRAVITATION
13-6 PLANETS AND SATELLITES: KEPLER’S LAWS
After reading this module, you should be able to . . .
13.17 Identify Kepler’s three laws.
13.18 Identify which of Kepler’s laws is equivalent to the law
of conservation of angular momentum.
13.19 On a sketch of an elliptical orbit, identify the semimajor
axis, the eccentricity, the perihelion, the aphelion, and the
focal points.
13.20 For an elliptical orbit, apply the relationships between
the semimajor axis, the eccentricity, the perihelion, and the
aphelion.
13.21 For an orbiting natural or artificial satellite, apply
Kepler’s relationship between the orbital period and radius
and the mass of the astronomical body being orbited.
● The motion of satellites, both natural and artificial, is governed by Kepler’s laws:
1. The law of orbits. All planets move in elliptical orbits with
the Sun at one focus.
2. The law of areas. A line joining any planet to the Sun
sweeps out equal areas in equal time intervals. (This
statement is equivalent to conservation of angular
momentum.)
3. The law of periods. The square of the period T of any
planet is proportional to the cube of the semimajor axis a
of its orbit. For circular orbits with radius r,
(law of periods),
where M is the mass of the attracting body—the Sun in the
case of the solar system. For elliptical planetary orbits, the
semimajor axis a is substituted for r.
T
2 ϭ
4
2
GM r
3
Learning Objectives
Key Ideas
tially at distance 10R E and finally at distance R E , where R E is
Earth’s radius (6.37 ϫ 10
6
m). Substituting Eq. 13-21 for U
and
for K, we rewrite Eq. 13-29 as
Rearranging and substituting known values, we find
ϭ (12 ϫ 10
3
m/s)
2
ϭ 2.567 ϫ 10
8
m
2
/s
2
,
and
v f ϭ 1.60 ϫ 10
4
m/s ϭ 16 km/s.
(Answer)
At this speed, the asteroid would not have to be particularly large to do considerable damage at impact. If it
were only 5 m across, the impact could release about as
much energy as the nuclear explosion at Hiroshima.
Alarmingly, about 500 million asteroids of this size are
near Earth’s orbit, and in 1994 one of them apparently
penetrated Earth’s atmosphere and exploded 20 km
above the South Pacific (setting off nuclear-explosion
warnings on six military satellites).
ϩ
2(6.67 ϫ 10
Ϫ11
m
3
/kgиs
2
)(5.98 ϫ 10
24
kg)
6.37 ϫ 10
6
m
0.9
v
2
f ϭ v
2
i ϩ
2GM
R E
1 Ϫ
1
10
1
2 mv
2
f Ϫ
GMm
R E
ϭ
1
2 mv
2
i Ϫ
GMm
10R E
.
1
2 mv
2
Sample Problem 13.03 Asteroid falling from space, mechanical energy
An asteroid, headed directly toward Earth, has a speed of
12 km/s relative to the planet when the asteroid is 10 Earth
radii from Earth’s center. Neglecting the effects of Earth’s
atmosphere on the asteroid, find the asteroid’s speed v f
when it reaches Earth’s surface.
KEY IDEAS
Because we are to neglect the effects of the atmosphere on
the asteroid, the mechanical energy of the asteroid – Earth
system is conserved during the fall. Thus, the final mechanical energy (when the asteroid reaches Earth’s surface) is
equal to the initial mechanical energy. With kinetic energy K
and gravitational potential energy U, we can write this as
K f ϩ U f ϭ K i ϩ U i .
( 1 3 - 2 9 )
Also, if we assume the system is isolated, the system’s linear momentum must be conserved during the fall. Therefore,
the momentum change of the asteroid and that of Earth must
be equal in magnitude and opposite in sign. However, because
Earth’s mass is so much greater than the asteroid’s mass, the
change in Earth’s speed is negligible relative to the change in
the asteroid’s speed. So, the change in Earth’s kinetic energy is
also negligible.Thus, we can assume that the kinetic energies in
Eq. 13-29 are those of the asteroid alone.
Calculations: Let m represent the asteroid’s mass and M
represent Earth’s mass (5.98 ϫ 10
24
kg). The asteroid is iniAdditional examples, video, and practice available at WileyPLUS