Question

The radii of orbits of two satellites revolving around the earth are in the ratio 3 : 8. Compare their (i) critical speeds (ii) period of revolution

Answer 1

Here $\frac{r_1}{r_2}=\frac{3}{8}$

By kepler's law, $T^2\alpha r^3$

$(\frac{T_1}{T_2}{)}^2=(\frac{r_1}{r_2}{)}^3$

$\frac{T_1}{T_2}=(\frac{r_1}{r_2}{)}^{3/2}=(\frac{3}{8}{)}^{3/2}$

$\frac{{\omega }_2}{{\omega }_1}=(\frac{3}{8}{)}^{3/2}$

$\frac{V_2{r}_2}{V_1{r}_1}=(\frac{3}{8}{)}^{3/2}$

$\frac{V_2}{V_1}=\left(\frac{r_1}{r_2}\right)(\frac{3}{8}{)}^{3/2}$

$=\left(\frac{3}{8}\right)\left({\frac{3}{8}}^{3/2}\right)$

$=(\frac{3}{8}{)}^{5/2}$

Ratio of speeds $\frac{V_2}{V_1}$ $=(\frac{3}{8}{)}^{5/2}$