Question

Eample 8.8 A 400 kg satellite is in a
circular orbit of radius 2R, about the
Earth. How much energy is required to
transfer it to a circular orbit of radius 4R?
What are the changes in the kinetic and
potential energies ?
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Answer 1

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Answer Text

Solution :

Here, m=400,

RE=6.37×106m

Initial total energy, Ei=PE+KE

=−GMmr+12mυ20=−GMmr+12m(GMr)

=−GMm2r=−GMm2×(2RE)=−GMm4RE

Final total energy, Ef=−GMm2×(4RE)=−GMm8RE

The change in total energy is, ΔE=Ef−Ei

−GMm8RE−(−GMm4RE)=GMm8RE

=(GMR2E)×mRE8=gmRE8

=(9.81)×(400)×(6.37×106)8

=3.13×109J Thus, teh energy required to transfer the satllite to the desired orbit =3.13×109J

the decrease in K.E.=Ki−Kf

=12m(GM2RE)−12m(GM4RE)

=18GMmRE=gmRE8=3.13×109J

Gain in P.E.=Uf−Ui

=−GMm4R−(−GMm2R)=GMm4R

=gmRE4=2×3.13×109J=6.26×109J

Thus, the again in P.E. is twice the loss in