Earth is moving around the sun in elliptical orbit as shown. The ratio of $O B$ and $O A$ is $R$. Then the ratio of Earth's velocities at $A$ and $B$ is

• A

  $R^{-1}$
• B

  $\sqrt{R}$
• Correct answer
  C

  $R$
• D

  $R^{2 / 3}$

Solution

In central force motion, the angular momentum of body remains conserved.

i.e., Angular momentum, $L=r \times p=m v r$ [at minor and major axis $\mathbf{v}$ and $\vec{r}$ are perpendicular] So, if $v_{1}$ and $v_{2}$ be the speed of earth at $A$ and $B$, respectively Then, $\quad m v_{1} r_{1}=m v_{2} r_{2}$ $\Rightarrow \quad \frac{v_{1}}{v_{2}}=\frac{r_{2}}{r_{1}}=\frac{O B}{O A} \Rightarrow \frac{v_{1}}{v_{2}}=R$ So, option (c) is correct.

Hence, the correct answer is option: 'C'.