Question

From a planet of mass $16M$ and radius $4R$, we wish to send an object to its moon of mass $M$ and radius $R$. Distance between centres of planet and moon is $10R$. Minimum speed at which object should be projected from the surface of planet so that it reaches moon is (Assume that the planet and the moon are at rest)

• A. $\sqrt{\frac{10GM}{3R}}$
• B. $\sqrt{\frac{8GM}{R}}$
• C. $\sqrt{\frac{7GM}{3R}}$
• D. $\sqrt{\frac{9GM}{2R}}$

Answer 1

The correct option is A $\sqrt{\frac{10GM}{3R}}$


The object has to at least reach the point where the force on it due to the planet and the moon cancels out and becomes zero.
Say, the distance between the planet and that point is $x_1$ and the distance between the moon and that point is $x_2$. So,
$\frac{G16Mm}{x_1^2}=\frac{GMm}{x_2^2}$
$\Rightarrow x_1=4x_2$
And,
$x_1+x_2=10R$
$\Rightarrow x_1=8R\text{and}{x}_2=2R$

Lets conserve the initial energy and the final energy at that point.

$\frac{1}{2}m{V}^2-\frac{G16Mm}{4R}-\frac{GMm}{(10R-4R)}=-\frac{G16Mm}{8R}-\frac{GMm}{2R}$
$\Rightarrow V=\sqrt{\frac{10GM}{3R}}$