Example 10-6 Escaping from a Martian Moon
You have landed your spacecraft on Phobos, the larger of the two airless moons
of Mars (Figure 10-10). Phobos is roughly spherical with an average radius of
11.1 km = 1.11 * 104 m. By dropping a baseball while standing on Phobos, you
find that g has a very small value at the moon’s surface, only 0.00580 m>s 2. At
what speed would you have to throw the baseball to have it escape Phobos?
Figure 10-10 ​Phobos What is the escape speed from this miniature moon of Mars, just 11.1 km
in radius?
Set Up
To find the escape speed vescape using Equation 10-10, we need the radius of Phobos,
which we are given, and the mass of Phobos,
which we are not given. However, we ­recall
from Equation 10-3 in Section 10-2 that the
value of g on Earth’s surface is related to
Earth’s mass and ­radius. We can write this same
equation for Phobos, which will let us use the
measured value of g on Phobos to determine its
mass.
Escape speed from Phobos:
vescape
2GmPhobos
=
B R Phobos
(10-10)
g = 0.00580 m/s2
Value of g at the surface of Phobos:
g =
GmPhobos
R 2Phobos
Phobos
radius 11.1 km
(10-3)
Solve
First use Equation 10-3 to determine the mass
of Phobos.
baseball
speed = ?
From Equation 10-3, g =
GmPhobos
R 2Phobos
Solve this for the mass of Phobos:
mPhobos =
10.00580 m>s 2 2 11.11 * 104 m2 2
gR 2Phobos
=
G
6.67 * 10-11 N # m2 >kg 2
= 1.07 * 1016
kg 2 # m>s 2
N
(Recall that 1 N = 1 kg # m>s 2.)
Then use Equation 10-10 to find the escape
speed from Phobos.
= 1.07 * 1016 kg
Escape speed:
vescape =
=
2GmPhobos
B R Phobos
C
216.67 * 10-11 N # m2 >kg 2 2 11.07 * 1016 kg2
1.10 * 104 m
kg # m m
N#m
m2
=
127
=
127 2
=
127 2
B
kg
B
kg
B
s
s
= 11.3 m>s = 40.7 km>h = 25.3 mi>h
(We once again used 1 N = 1 kg # m>s 2.)
Reflect
The escape speed on Phobos is almost exactly 1>1000 the escape speed on Earth! In general, the smaller the planet or
moon, the lower the escape speed.
A typical high school baseball pitcher can easily throw a baseball at more than 25 m>s (90 km>h or 56 mi>h), so a
well-trained astronaut should have no problem throwing a baseball at the 11.3@m>s escape speed on Phobos.