# Speed of a planet in an elliptical orbit with semi major axis 'a' about sum of mass M at a distance r from the sun is

$(a)\;\sqrt {GM \bigg(\frac{2}{r}-\frac{1}{a}\bigg)} \quad (b)\;\sqrt {GM \bigg(\frac{1}{r}-\frac{1}{a}\bigg)} \quad (c)\;\sqrt {GM \bigg(\frac{1}{r}-\frac{2}{a}\bigg)} \quad (d)\;\sqrt {\frac{GMr}{2a^2}}$

Total energy of a planet in an elliptical orbit is
$E= \large\frac{-GMm}{2a}$ (m= mass of planet )
By conservation of mechanical energy
$KE+P.E= E$
$\large\frac{1}{2}$$mv^2 -\large\frac{GMm}{r}=\frac{-GMm}{2a}$
$v=\sqrt {GM \bigg(\large\frac{2}{r}-\frac{1}{a}\bigg)}$
Hence a is the correct answer
edited Jul 3, 2014