Question

If a satellite is moving around a planet of mas M in a elliptical orbit of semi major axis a.The orbital speed (V) of the satellite when it is at a distance r from the focus will be given by

Answer 1

Conservation of energy implies that sum of KE and gravitational potential for the satellite is also conserved along the path. The total energy of the satellite in elliptical path in $\frac{-GMm}{2a}$

G is Gravitational constant

M$=$ mass of p level

m$=$ mass of satellite

$\therefore$ The total energy $=K.E+P.E$

$\frac{1}{2}m{v}^2-\frac{GMm}{r}=-\frac{GMm}{2a}$

$\frac{1}{2}m{v}^2=\frac{GMm}{r}-\frac{GMm}{2a}$

$v^2=GM(\frac{2}{r}-\frac{1}{a})$

$v=\sqrt{GM(\frac{2}{r}-\frac{1}{a})}$.