A satellite is put in an orbit just above the ear

Question

A satellite is put in an orbit just above the earth's atmosphere with a velocity (√1.5) times the velocity for a circular orbit at that height. The initial velocity imparted in horizontal. What would be the maximum distance of the satellite from the surface of the earth when it is in orbit

Arun · Accepted Answer

Dear Rajesh

From Energy conservation we get,
1/2 mv1^2 - GMm/r1 = 1/2 mv2^2 - GMm/r2 …...........(1)

and momentum conservation as no external forces are acting,
m r1 v1 = m r2 v2 ….............(2)
From Newton’s second law of equation we get,
1/2 v1^2 - GM/r1 = 1/2 (r1 v1/r2)^2 - GM/r2 ….............(3)
From eqns. (1) (2) and (3) on solving we get,
(v1^2 - 2GM/r1) r2^2 = (r1 v1)^2 - 2GMr2
Given, velocity is sqrt(1.5) times the velocity of the circular orbit at the initial radius.

Hence,
GMm/r1^2 = mv^2/r1

or
v1^2 = 1.5GM/r1
Furthet solving,
(-0.5GM/r1) r2^2 = 1.5GMr1 - 2GMr2
r2^2 - 4r1 r2 + 3r1^2 = 0
and hence solving the quadratic,
r2 = 2r1 +- sqrt(16r1^2 - 12r1^2)/2
r2 = 2r1 +- r1
r2 = r1, 3r1

Let R = radius of the earth,

B0 = initial altitude of the orbit,

A = maximum altitude.

Hence,
R+A = 3(R+B0)

or
A = 2R + 3B0
On putting earth’s radii and B0 value we get

A = 18742 km.

Regards

Arun (askIITians forum expert)

Mechanics> A satellite is put in an orbit just above...

A satellite is put in an orbit just above the earth's atmosphere with a velocity (√1.5) times the velocity for a circular orbit at that height. The initial velocity imparted in horizontal. What would be the maximum distance of the satellite from the surface of the earth when it is in orbit

Rajesh ganesan , 7 Years ago

Arun

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Mechanics> A satellite is put in an orbit just above...

A satellite is put in an orbit just above the earth's atmosphere with a velocity (√1.5) times the velocity for a circular orbit at that height. The initial velocity imparted in horizontal. What would be the maximum distance of the satellite from the surface of the earth when it is in orbit

Rajesh ganesan , 7 Years ago

Arun

LIVE ONLINE CLASSES

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