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Solution 1
Conservation of energy given by the sum of potential energy due to gravity and kinetic
energy can be used to determine escape velocity.
In the case of Earth along the potential
is given by:
r
m
M
G
r
E
−
=
)
(
φ
where m is the mass of the book. The book will escape if initial kinetic energy is high
enough to overcome the potential at
E
R
r
=
. Thus
E
E
E
R
m
M
G
mv
=
2
2
thus
s
km
R
GM
v
E
E
E
/
11
2
=
=
In the Earth-Moon case the potential is
r
R
m
M
G
r
m
M
G
r
EM
M
E
−
−
−
=
)
(
where MM = ME. The potential is a symmetric double-well and in order to leave the
surface of the earth the kinetic energy must be high enough to overcome a saddle point
right in the middle between earth and moon. Thus the condition for escape velocity is
ME
E
E
EM
E
E
E
R
m
GM
R
R
R
m
GM
mv
4
1
1
2
2
−
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
+
−
This equation solved for escape velocity gives
.
/
7
.
7
s
km
v
E
=
Solution 2
Introduce the generalized coordinates as in the figure below.
and

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