Question

A planet orbits the sun in an elliptical path as shown in the figure. Let $v_P$ and $v_A$ be speed of the planet when at perihelion and aphelion respectively. Which of the following relations is correct?

• A. $\frac{r_P}{r_A}=\frac{v_A}{v_P}$
• B. $\frac{r_P}{r_A}=\frac{v_P}{v_A}$
• C. $\frac{r_P}{r_A}=\sqrt{\frac{v_P}{v_A}}$
• D. $\frac{r_P}{r_A}=\sqrt{\frac{v_A}{v_P}}$

Answer 1

Answer: (A)

Angular momentum of planet at $P$, $L_P=m_p{v}_p{r}_p$
where $m_p$ is the mass of the planet.
Angular momentum of planet at $A$,
$L_A=m_p{v}_A{r}_A$
According to the law of conservation of angular momentum
$L_P=L_A$, $m_p{v}_p{r}_p=m_p{v}_A{r}_A$
or $\frac{r_P}{r_A}=\frac{v_A}{v_P}$