Question

Find the velocity of a satellite travelling in an elliptical orbit, when it reaches point $\text{C}$ on the end of the semi-minor axis.

• A. $\sqrt{\frac{GM}{a}}$
  
• B. $\sqrt{\frac{GM}{2a}}$
  
• C. $\sqrt{\frac{2GM}{a}}$
  
• D. $\sqrt{\frac{GM}{3a}}$
  

Answer 1

The correct option is A $\sqrt{\frac{GM}{a}}$



From the property of ellipse we know that
$r=a$ (semi major axis)

Also, the total energy of a particle in elliptical orbit is given by

$E_1=\frac{-GMm}{2a}$

Let $v$ be the speed of the particle at $\text{C}$. Then the total energy at this point will be,

$E_2=\frac{1}{2}m{v}^2+\frac{-GMm}{r}$

From the conservation of energy,

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

$\Rightarrow v=\sqrt{\frac{GM}{a}}$

Hence, the option (a) is the correct answer.

Why this question: It is important to remember that the total energy of a particle in elliptical orbit is $$E=\frac{-GMm}{2a}$$ Note: From the property of ellipse $(a>b)$, $b^2=a^2(1-e^2)$ $\Rightarrow r^2=b^2+(ae{)}^2=a^2$