Question

If a satellite is revolving around a planet of mass $M$ in an elliptical orbit of semi-major axis $a$ , find the orbital speed of the satellite when it is at a distance $r$ from the focus

• A. $v^{2}=GM\left[\frac{2}{r} - \frac{1}{a}\right]$
• B. $v^{2}=GM\left[\frac{2}{r^{2}} - \frac{1}{a}\right]$
• C. $v^{2}=GM\left[\frac{2}{r^{2}} - \frac{1}{a^{2}}\right]$
• D. $v^{2}=G\left[\frac{2}{r} - \frac{1}{a}\right]$

Answer 1

Answer: (A)

As in case of elliptic orbit of a satellite mechanical energy
remains constant, at any position of satellite in the orbit,
$KE \, + \, PE \, =-\frac{G M m}{2 a}$ ...(i)
Now, if at position $r,v$ is the orbital speed of satellite
$KE \, = \, \frac{1}{2} \, mv^{2} \, $ and $PE \, = \, -\frac{G M m}{r}$ ...(ii)
So, from Eqs. (i) and (ii), we have
$\frac{1}{2} \, mv^{2}-\frac{G M m}{r}=-\frac{G M m}{2 a}, \, i.e., \, v^{2}=GM\left[\frac{2}{r} - \frac{1}{a} \, \right]$