An artificial satellite (of mass m) of the earth (radius R and mass M) moves in an orbit whose radius in n times the radius of the earth. Assuming resistance to the motion to be proportional to e square velocity, that is, F=av2 , find how long the satellite will take to fall onn to the earth.
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2 Answers
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1orbital speed at a distance 'r' from the centre of the earth is given by v=\sqrt{\frac{GM}{r}}
K.E. = 12mv2 = GMm2r
P.E. = - GMmr
total mechanical energy = K.E.+P.E. = - GMm2r
now dEdt = - F.v
or GMm2r2(drdt) = - av3
GMm2r2(drdt) = - a (GMr)\sqrt{\frac{GM}{r}}
\frac{a}{m}\int_{0}^{t}{dt}=-\frac{1}{2}\sqrt{\frac{1}{GM}}\int_{r_{i}}^{r_{f}}{r^{-1/2}}dr
\frac{a}{m}t=-\frac{1}{2}\sqrt{\frac{1}{GM}}\int_{nR}^{R}{r^{-1/2}}dr
t = \frac{m}{a}\sqrt{\frac{R}{GM}}(\sqrt{n}-1)
hence the answer [1]