Question

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

• A. $\sqrt{GM[\frac{2}{r}-\frac{1}{a}]}$
• B. $\sqrt{GM[\frac{1}{r}-\frac{1}{a}]}$
• C. $\sqrt{GM[\frac{1}{r}-\frac{2}{a}]}$
• D. $\sqrt{\frac{GMr}{2a^2}}$

Answer 1

The correct option is A $\sqrt{GM[\frac{2}{r}-\frac{1}{a}]}$

Total energy of planet in elliptical orbits

$E=\frac{-{GM}_m}{2a}$ ($M=$ mass of planet)

By conservation of mechanical energy

$k\epsilon +p\epsilon =\epsilon$

$\frac{1}{2}{mV}^2-\frac{{GM}_m}{r}=\frac{-{GM}_m}{2a}$

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

Hence $\left(A\right)$ is correct answer.