Question

The minimum and maximum distances of a planet revolving around sun are $r$ and $R$. If the minimum speed of planet on its trajectory is $v_0$, its maximum speed will be

• A. $\frac{v_{0}R}{r}$
• B. $\frac{v_{0}r}{R}$
• C. $\frac{v_0{R}^2}{r^2}$
• D. $\frac{v_0{r}^2}{R^2}$

Answer 1

The correct option is A $\frac{v_{0}R}{r}$

Given,

minimum velocity of planet, $V_{min}=v_0$,

Let, maximum velocity of planet, $V_{max}$.


Accoring to Kepler's second law the line that joins any planet to the sun sweeps out equal areas in equal intervals of time. Thus planets appear to move slower when they are farther sun than when they are nearer.

Now, for planets moving around the in an elliptical orbit, angular momentum is conserved.

$\Rightarrow \left|\begin{array}{c}{\overline{L}}_P\end{array}\right|=\left|\begin{array}{c}{\overline{L}}_A\end{array}\right|$

$\Rightarrow m{V}_{max}r=m{v}_{0}R$

$\therefore V_{max}=\frac{v_{0}R}{r}$

Why this question: To make students understand the consequence of conservation of angular momentum in planetary orbits. The orbital velocity will be greater when the planet is closer to the sun. Also when velocity at one point is known, velocity at another point can be found out.