Question

A planet revolves about the sun in elliptical orbit. The arial velocity $\left(\frac{dA}{dt}\right)$ of the planet is $4.0\times 10^{16}{m}^2/s$. The least distance between planet and the sun is $2\times 10^{12}m$. Then the maximum speed of the planet in $km/s$ is:

• A. 10
• B. 20
• C. 40
• D. None of these

Answer 1

Answer: (C)

$\frac{dA}{dt}=\frac{r^2\omega }{2}$ is constant
$\therefore \frac{dA}{dt}=\frac{r_{\max }^2{\omega }_{\min }}{2}=\frac{r_{\min }^2{\omega }_{\max }}{2}$
$\Rightarrow {\omega }_{\min }=\frac{2dA/dt}{r_{\max }^2}$
$V_{\max }={\omega }_{\min }{r}_{\min }=\frac{2dA/dt}{r_{\min }}=40km/s$