Question

A planet is revolving in an elliptical orbit around the Sun. Its closest distance from the Sun is $r_{min}$ and the farthest distance is $r_{max}$. If the velocity of the planet at the distance of the closest approach is $v_1$ and that at the farthest distance from the Sun is $v_2$, then $v_1/v_2$.

• A. $\frac{r_{max}}{r_{min}}$
• B. $\frac{r_{min}}{r_{max}}$
• C. $\frac{r_{min}+r_{max}}{r_{max}-r_{min}}$
• D. None

Answer 1

The correct option is A $\frac{r_{max}}{r_{min}}$

Since angular momentum is conserved therefore

$I_1{w}_1=I_2{w}_2$

or ${MR}^2{w}_1={MR}^2{w}_2$

But $w_1=\frac{V_1}{r_{min}}$

$w_2=\frac{V_2}{r_{max}}$

$\therefore$ $r_{min}^2\times \frac{V_1}{r_{min}}=r_{max}^2\times \frac{V_2}{r_{max}}$

$V_1{r}_{min}=V_2{r}_{max}$

$V_2=\frac{V_1{r}_{min}}{r_{max}}$

$\frac{V_2}{V_1}=\frac{r_{min}}{r_{max}}$

and $\frac{V_1}{V_2}=\frac{r_{max}}{r_{min}}$

$\frac{V_1}{V_2}=\frac{r_{max}}{r_{min}}$