Question

A satellite is in an ellipitcal orbit around the earth with aphelion of $5R$ and perihelion $3R$ where $R$ is the radius of earth. At the instant it is passing through the apogee, by what amount its speed should be increased in order to make its orbit circular? (M = mass of earth)

• A. $\sqrt{\frac{GM}{R}}\left[\frac{3-\sqrt{3}}{5}\right]$
• B. $\sqrt{\frac{GM}{R}}\left[\frac{2-\sqrt{3}}{\sqrt{20}}\right]$
• C. $\sqrt{\frac{GM}{R}}\left[\frac{2-\sqrt{2}}{\sqrt{20}}\right]$
• D. $\sqrt{\frac{GM}{R}}\left[\frac{2-\sqrt{2}}{5}\right]$

Answer 1

The correct option is B $\sqrt{\frac{GM}{R}}\left[\frac{2-\sqrt{3}}{\sqrt{20}}\right]$

$V_1^{5}R=V_{2}3R$

$5V_1=3V_2\Rightarrow V_2=\frac{5V_1}{3}$

$-\frac{GMm}{5R}+\frac{1}{2}m{v}_1^2=\frac{GMm}{3R}+\frac{1}{2}m\cdot \frac{25}{9}{V}_1^2$

$\Rightarrow \frac{1}{2}{V}_1^2[\frac{25}{9}-1]=\frac{GM}{R}[\frac{1}{3}-\frac{1}{5}]=\frac{GM}{R}\left(\frac{5-3}{15}\right)$

$V_1=\sqrt{\frac{2GM}{15r}}\times \frac{2\times 9}{16}=\sqrt{\frac{3GM}{20R}}$

$\frac{V_c^2}{5R}=\frac{GM}{25R^2}$

$\Rightarrow V_C=\sqrt{\frac{GM}{5R}}$

$\Delta V=V_c-V_1=\sqrt{\frac{GM}{R}}[\frac{1}{\sqrt{5}}-\sqrt{\frac{3}{20}}]$

$=\sqrt{\frac{GM}{R}}\left[\frac{2-\sqrt{3}}{\sqrt{20}}\right]$