Question

The ratio of kinetic energy of a planet at perigee and apogee during its motion around the sun in elliptical orbit of eccentricity e is

• A. $1:e$
• B. $\frac{1+e}{1-e}$
• C. ${\left(\frac{1+e}{1-e}\right)}^2$
• D. ${\left(\frac{1-e}{1+e}\right)}^2$

Answer 1

Answer: (C)

$K.E$ of a planet $=\frac{1}{2}m{v}^2$
$K.E$ at perigee $=\frac{1}{2}m{v}_p^2$
$K.E$ at apogee $=\frac{1}{2}m{v}_A^2$
Using conservation of angular momentum at $P$ and $A$.
$\Rightarrow m{v}_p{r}_p=m{v}_A{r}_A$
$\Rightarrow \frac{v_P}{v_A}=\frac{r_A}{r_P}=\frac{a(1+e)}{a(1-e)}$
$\Rightarrow \frac{K.E_P}{K.E_A}=\frac{v_P^2}{v_A^2}={\left(\frac{1+e}{1-e}\right)}^2$