Question
A satellite of mass $$m$$ is orbiting the earth in a circular orbit of radius $$R.$$ It starts losing energy due to small air resistance at the rate of $$C\,J/s.$$ Find the time taken for the satellite to reach the earth.
A.
$$\frac{{GMm}}{C}\left[ {\frac{1}{R} - \frac{1}{r}} \right]$$
B.
$$\frac{{GMm}}{{2C}}\left[ {\frac{1}{R} + \frac{1}{r}} \right]$$
C.
$$\frac{{GMm}}{{2C}}\left[ {\frac{1}{R} - \frac{1}{r}} \right]$$
D.
$$\frac{{2GMm}}{C}\left[ {\frac{1}{R} + \frac{1}{r}} \right]$$
Answer :
$$\frac{{GMm}}{{2C}}\left[ {\frac{1}{R} - \frac{1}{r}} \right]$$
Solution :
$$\eqalign{
& E = - \frac{{GMm}}{{2r}} \cr
& - \frac{{dE}}{{dt}} = \frac{{GMm}}{{2r}}\frac{1}{{{r^2}}}\frac{{dr}}{{dt}} \cr
& \int\limits_0^t {dt} = - \frac{{GMm}}{{2C}}\int\limits_r^R {\frac{{dr}}{{{r^2}}}} ;t = \frac{{GMm}}{{2C}}\left[ {\frac{1}{R} - \frac{1}{r}} \right] \cr} $$