Question

Let the speed of the planet at the perihelion $P$ be $v_{p}$ and the sun-planet distance be $r_{p} .$ Relate $\left(r_{p}, v_{p}\right)$ to the corresponding quantities at the aphelion $\left(r_{a}, v_{a}\right)$

• A. $\frac{r_{p}}{r_{a}}=\frac{v_{p}}{v_{a}}$
• B. $\frac{r_{p}^{2}}{r_{a}^{2}}=\frac{v_{a}}{v_{p}}$
• C. $\frac{v_{p}}{v_{a}}=\frac{r_{a}}{r_{p}}$
• D. $\frac{v_{p}^{2}}{v_{a}^{2}}=\frac{r_{a}}{r_{p}}$

Answer 1

Answer: (C)

At the perihelion $r_{p}$ and $v_{p}$ are mutually perpendicular so $r_{a}$ is normal to $v_{a}^{p}$. The angular momentum
is conserved,
$\therefore m v_{a} r_{a}=m v_{p} r_{p} \Rightarrow \frac{v_{p}}{v_{a}}=\frac{r_{a}}{r_{p}}$