5.3 Solutions
217
Fig. 5.16
At the apogee (farthest point), the velocity of the satellite is perpendicular to the radius vector. Therefore, the angular momentum at the apogee
= (mv)(5R/2).
Conservation of angular momentum gives
5
2
mv R =
m
2
v 0 R
or v =
v 0
5
(1)
The kinetic energy at the surface K 0 =
1
2
mv 2
0 and potential energy U 0 =
−
G Mm
R
.
Therefore, the total mechanical energy at the surface is
E 0 =
1
2
mv
2
0 −
G Mm
R
(2)
At the apogee kinetic energy K =
1
2
mv 2 and the potential energy U =
−
2G Mm
5R
.
Therefore, the total mechanical energy at the apogee is
E =
1
2
mv
2
−
2
5
G Mm
R
(3)
Conservation of total energy requires that E = E 0 . Eliminating v in (3) with
the aid of (1) and simplifying we get
v 0 =
5G M
4R