Question

A planet orbits in an elliptical path of eccentricity $e$ around a massive star considered fixed at one of the foci. The point in space where it is closest to the star is denoted by $P$ and the point where it is farthest is denoted by $A$. Let $v_P$ and $v_A$ be the respective speeds at $P$ and $A$. Then

• A. $\frac{V_P}{V_A}=\frac{1+e}{1-e}$
• B. $\frac{V_P}{V_A}=1$
• C. $\frac{V_P}{V_A}=\frac{1+e^2}{1-e}$
• D. $\frac{V_P}{V_A}=\frac{1+e^2}{1-e^2}$

Answer 1

The correct option is A $\frac{V_P}{V_A}=\frac{1+e}{1-e}$

Let the larger radius be $r_A=a(1+e)$ and $r_P=a(1-e)$

also angular momentum is conserved and $\theta ={90}^o$

So $m{v}_A{r}_A=m{v}_P{r}_P$

so $\frac{v_P}{v_A}=\frac{1+e}{1-e}$