Question

A stationary object is released from a point $P$ a distance $3R$ from the centre of the moon which has radius $R$ and mass $M$ . Which one of the following expressions gives the speed of the object on hitting the moon?

• A. $\left(\frac{2 G M}{3 R}\right)^{\frac{1}{2}}$
• B. $\left(\frac{4 G M}{3 R}\right)^{\frac{1}{2}}$
• C. $\left(\frac{2 G M}{R}\right)^{\frac{1}{2}}$
• D. $\left(\frac{G M}{R}\right)^{\frac{1}{2}}$

Answer 1

Answer: (B)

According to the law of conservation of energy,
$-\frac{G m M}{3 R}+\frac{m v_{0}^{2}}{2}=-\frac{G m M}{R}+\frac{m v_{1}^{2}}{2}$
Where $m$ is the mass of the object, $v_{0}$ and $v_{1}$ are the initial and final velocities of object, respectively.
The object is stationary i.e., $v_{0}=0.$
$\Rightarrow \, -\frac{G M}{3 R}=-\frac{G M}{R}+\frac{v_{1}^{2}}{2}$
$\Rightarrow \, \, \frac{v_{1}^{2}}{2}=\frac{2 G M}{3 R}$
$\Rightarrow \, \, v_{1}=2\sqrt{\frac{G M}{3 R}}=\left(\frac{4 G M}{3 R}\right)^{\frac{1}{2}}$