Solve Study Textbooks Guides

Join / Login

Question

The masses and radii of the earth and moon are $M_{1}$ , $R_{1}$ and $M_{2}$ , $R_{2}$ respectively. Their centres are at distance $d$ apart. The minimum velocity with which a particle of mass $m$ should be projected from a point midway between their centres so that it escapes to infinity is:

A

$0$

B

$2 \frac{G m}{d} (M_{1} + M_{2})$

C

$2 \frac{G}{d} (M_{1} + M_{2})$

D

$2 \frac{G}{m d} (M_{1} + M_{2})$

Hard

Open in App

Solution

Verified by Toppr

Correct option is C)

The sum total of kinetic energy and potential energy of system must be conserved. For minimum velocity, this must be $0$ at infinity.

Hence,

$\frac{1}{2} m v^{2} - \frac{G M _{1} m}{\frac{d}{2}} - \frac{G M _{2} m}{\frac{d}{2}} = 0$

$\frac{1}{2} m v^{2} = 2 \frac{G m}{d} (M_{1} + M_{2})$

$v = 2 \frac{G}{d} (M_{1} + M_{2})$

Was this answer helpful?

0

0

Similar questions

If 'd' is the distance between the centers of the earth of mass $M_{1}$ and moon of mass $M_{2}$ , then the velocity with which a body should be projected from the mid point of the line joining the earth and the moon, so that it just escapes is :

Masses and radii of earth and moon are $M_{1}, M_{2}$ and $R_{1}, R_{2}$ respectively. The distance between theircentre is 'd'. The minimum velocity given to mass 'M' from the mid point of line joining their centre so that it will escape.

The masses and radii of the earth and the Moon are $M_{1}$ , $R_{1}$ and $M_{2}$ , $R_{2}$ , respectively. Their centres are at distance $d$ apart. The minimum speed with which a particle of $m$ should be projected from a point midway the two centres so as to escape to infinity is:

Three planets of same density have radii $R_{1}, R_{2}$ and $R_{3}$ such that $R_{1} = 2 R_{2} = 3 R_{3}$ The gravitational field at their respective surfaces are $g_{1}, g_{2}$ and $g_{3}$ and escape velocities from their surfaces are $v_{1}, v_{2}$ and $v_{3}$ then

This question has multiple correct options

The masses and radii of the earth and the Moon are $M_{1}$ , $R_{1}$ and $M_{2}, R_{2}$ respectively. Their centres are a distance $d$ apart. The minimum speed with which a particle of mass $m$ should be projected from a point midway between teh two centres so as to escape to infinity is $n \frac{G ( M _{1} + M _{2} )}{d}$ . The value of $n$ is:

View more