4th Asian Physics Olympiad Theoretical Competition 23 April 2003
Theoretical Competition
I. Satellite’s orbit transfer
In the near future we ourselves may take part in launching of a satellite which, in
point of view of physics, requires only the use of simple mechanics.
u
2
P
•
R
1
u
0
M•
•
m
R
• 0
Q
u
1
a) A satellite of mass mis presently circling the Earth of mass M in a circular orbit of
radiusR . What is the speed (u )of mass m in terms of M,R and the universal
0 0 0
gravitation constantG? (1 point)
b) We are to put this satellite into a trajectory that will take it to point P at distance R
1
from the centre of the Earth by increasing (almost instantaneously) its velocity at
point Q from u to u . What is the value of u in terms of u ,R ,R ?
0 1 1 0 0 1
(2 points)
c) Deduce the minimum value of u in term of u that will allow the satellite to leave
1 0
the Earth’s influence completely. (1 point)
d) (Referring to part b.) What is the velocity (u )of the satellite at point P in terms of
2
u ,R ,R ? (1 point)
0 0 1
e) Now, we want to change the orbit of the satellite at point P into a circular orbit of
radius R by raising the value of u (almost instantaneously) to u .
1 2 3
What is the magnitude of u in terms of u ,R ,R ? (1 point)
3 2 0 1
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4th Asian Physics Olympiad Theoretical Competition 23 April 2003
f)
Y
•m
r
mean orbit of • θ X
M
radius R
1
If the satellite is slightly and instantaneously perturbed in the radial direction so that
it deviates from its previously perfectly circular orbit of radius R , derive the period
1
of its oscillation T of rabout the mean distance R .
1
Hint: Students may make use (if necessary) of the equation of motion of a satellite
in orbit:
d2 d 2 Mm
m r− θ r = −G ………………………… (1)
dt2 dt r2
and the conservation of angular momentum:
d
mr2 θ = constant ……………………….… (2)
dt
(3 points)
g) Give a rough sketch of the whole perturbed orbit together with the unperturbed one.
(1 point)
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