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 $P$, $L_P = m_pv_pr_p$
where $m_p$ is the mass of the planet.
Angular momentum of planet at $A$,
$L_A = m_pv_Ar_A$
According to the law of conservation of angular momentum
$L_P = L_A$, $m_pv_pr_p = m_pv_Ar_A$
or $\frac{r_{P}}{r_{A}} = \frac{v_{A}}{v_{P}}$