Two identical spherically symmetric planets, each of mass M, are somehow held at rest with respect to each other. Each planet has radius R, and the distance between the centers of the planets is 4R. If a rocket is launched from the surface of one planet with speed v, what is the minimum speed v so that the rocket can reach the other planet?

see full answer

Start JEE / NEET / Foundation preparation at rupees 99/day !!

21% of IItians & 23% of AIIMS delhi doctors are from Sri Chaitanya institute !!

An Intiative by Sri Chaitanya

$\sqrt{\frac{2 G M}{R}}$

$\sqrt{\frac{G M}{R}}$

$\sqrt{\frac{3 G M}{4 R}}$

$\sqrt{\frac{2 G M}{3 R}}$

answer is D.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

The gravitational force vanishes at the midway point between
the planets, so the rocket only needs to
have enough energy to get there. The initial and final gravitational potential energies are
$U_{i} = - \frac{G M m}{R} - \frac{G M m}{3 R} = - \frac{4 G M m}{3 R}$ an $U_{f} = - \frac{2 G M m}{2 R} = - \frac{G M m}{R}$

Thus, the initial kinetic energy needed is
$\frac{1}{2} m v^{2} = U_{f} - U_{i} = \frac{G M m}{3 R}$
which implies $v = \sqrt{\frac{2 G M}{3 R}}$

Watch 3-min video & get full concept clarity

Result Producing online coaching for JEE/NEET of south India

Largest All India Test Series for JEE/NEET

Best Books for IIT JEE

Best Books for NEET

How to Join IIT after 10th

Get Expert Academic Guidance – Connect with a Counselor Today!

Result Producing online coaching for JEE/NEET of south India

Largest All India Test Series for JEE/NEET

Best Books for IIT JEE

Best Books for NEET

How to Join IIT after 10th