Question

A spaceship approaches the Moon $($mass $=M$ and radius $=R)$ along a parabolic path which is almost tangential to its surface. At the moment of the maximum approach, the brake rocket is fired to convert the spaceship into a satellite of the Moon. The change in speed is

• A. $\sqrt{\frac{GM}{R}}(\sqrt{2}-1)$
• B. $\sqrt{\frac{GM}{R}}(\sqrt{3}-1)$
• C. $\sqrt{\frac{GM}{R}}(\sqrt{5}-1)$
• D. $\text{None of these}$

Answer 1

The correct option is A $\sqrt{\frac{GM}{R}}(\sqrt{2}-1)$

$v_{\text{orbit}}=\sqrt{\frac{GM}{R}}$
If $v$ is the velocity at the vertex of the parabola, then $v$ is also the escape velocity because if it is thrown with this velocity it will follow the parabolic path never to return to the moon.Now
$v_{escape}=\sqrt{\frac{GM}{R}}$
$△v=v_{final}-v_{initial}=v_{orbital}-v_{escape}$
$=\sqrt{\frac{GM}{R}}-\sqrt{\frac{2GM}{R}}=-\sqrt{\frac{GM}{R}}(\sqrt{2}-1)$
The negative sign means the speed has to be decreased.
$\therefore$ Required change in speed $=\sqrt{\frac{GM}{R}}(\sqrt{2}-1)$