Question

A satellite under the influence of the gravitational field of a massive planet at point $P$ moves along a parabolic path $y=x^2$ as shown. At point $O$, the satellite moves with a speed $v$. What is the time taken by satellite to move from point $O$ to $B?$

• A. $\frac{R}{3v}$
• B. $\frac{R\sqrt{R}}{3v}$
• C. $\frac{4\sqrt{R}}{3v}$
• D. $\frac{2R}{3v}$

Answer 1

The correct option is C $\frac{4\sqrt{R}}{3v}$

From Kepler's second law

Areal velocity, $\frac{\Delta A}{\Delta t}=\frac{L}{2m}=\frac{mvR}{2m}=\frac{vR}{2}$

Co-ordinate of $O$ and $B$ are $(0,0)$ and $(\sqrt{R},R)$, respectively.

Area swept by the line joining satellite and the planet from $O$ to $B$ is

$\Delta A=R\sqrt{R}-{\int }_0^{\sqrt{R}}{x}^{2}dx$

$\Delta A=R\sqrt{R}-\frac{R\sqrt{R}}{3}=\frac{2R\sqrt{R}}{3}$

$\therefore$ Time taken to go from $O$ to $B$,

$\Delta t=\frac{\Delta A}{\Delta A/\Delta t}=\frac{\frac{2R\sqrt{R}}{3}}{\frac{vR}{2}}=\frac{4\sqrt{R}}{3v}$

Hence, option $\left(\text{C}\right)$ is correct.