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

Answer 1

Answer: (B)

According to the law of conservation of energy,
$-\frac{GmM}{3R}+\frac{m{v}_0^2}{2}=-\frac{GmM}{R}+\frac{m{v}_1^2}{2}$
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., $v_0=0.$
$\Rightarrow -\frac{GM}{3R}=-\frac{GM}{R}+\frac{v_1^2}{2}$
$\Rightarrow \frac{v_1^2}{2}=\frac{2GM}{3R}$
$\Rightarrow v_1=2\sqrt{\frac{GM}{3R}}={\left(\frac{4GM}{3R}\right)}^{\frac{1}{2}}$