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

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) v2=GM[(2/r) - (1/a)] (B) v2=GM[(2/r2) - (1/a)] (C) v2=GM[(2/r2) - (1/a2)] (D) v2=G[(2/r) - (1/a)]

Tardigrade Admin · Accepted Answer

As in case of elliptic orbit of a satellite mechanical energy
remains constant, at any position of satellite in the orbit,

K E + PE = - \frac{GM m}{2 a}

...(i)
Now, if at position

r, v

is the orbital speed of satellite

K E = \frac{1}{2} m v^{2}

and

PE = - \frac{GM m}{r}

...(ii)
So, from Eqs. (i) and (ii), we have

\frac{1}{2} m v^{2} - \frac{GM m}{r} = - \frac{GM m}{2 a}, i . e ., v^{2} = GM [\frac{2}{r} - \frac{1}{a}]

Q. 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

$v^{2} = GM [\frac{2}{r} - \frac{1}{a}]$

$v^{2} = GM [\frac{2}{r ^{2}} - \frac{1}{a}]$

$v^{2} = GM [\frac{2}{r ^{2}} - \frac{1}{a ^{2}}]$

$v^{2} = G [\frac{2}{r} - \frac{1}{a}]$

Q. 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

v2=GM[r2​−a1​]

v2=GM[r22​−a1​]

v2=GM[r22​−a21​]

v2=G[r2​−a1​]

Q. 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

$v^{2} = GM [\frac{2}{r} - \frac{1}{a}]$

$v^{2} = GM [\frac{2}{r ^{2}} - \frac{1}{a}]$

$v^{2} = GM [\frac{2}{r ^{2}} - \frac{1}{a ^{2}}]$

$v^{2} = G [\frac{2}{r} - \frac{1}{a}]$