Question

Consider a 1000 kg communication satellite that needs to be boosted from an orbit 330 km above the earth to a geosynchronous orbit 35,900 km above the earth.

c) How much work must the rocket motor do to transfer the satellite from the circular orbit to the elliptical orbit?

e) Now find the velocity v2 of the outer circular orbit.

f) How much work must the rocket motor do to transfer the satellite from the elliptical orbit to the outer circular orbit?

g) Compute the total work done.

Answer 1

(c)

Let radius of earth be 6370 km

W = GMm / 2 ( 1 / r1 - 1/r2)

W = 6.67e-11 * 5.98e24 * 1000 / 2 ( 1 / 6700e3 - 1/42270e3 )

W = 2.5e10 J

______________________

(e)

v2 = sqrt ( GM / r)

v2 = sqrt ( 6.67e-11 * 5.98e24 / 42270e3)

v2 = 3.07e3 m/s

______________________

(f)

velocity in elliptical orbit

v2' = sqrt ( 2GM (R / r) / (r + R)

where

r = 42270e3 m

R = 6700e3 m

so,

v' = 1.6e3 m/s

so,

W = change in K.E

W = 1/2 * 1000 * ( 3.07e3 ² - 1.6e3²)

W = 3.42e9 J

_____________________________

(g)

total work done

W = 2.5e10 + 3.42e9

W = 2.84e10 J