(B) The potential energy $(PE)$ of a particle of mass $m$ placed at a point midway between the Earth and the Moon is given by:$U = -\left(\frac{GM_1 m

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$v = \sqrt {\frac{{4G({M_1} + {M_2})}}{d}} $

(B) The potential energy $(PE)$ of a particle of mass $m$ placed at a point midway between the Earth and the Moon is given by:$U = -\left(\frac{GM_1 m}{d/2} + \frac{GM_2 m}{d/2}\right) = -\frac{2Gm(M_1 + M_2)}{d}$The particle will escape to infinity if its total energy becomes at least zero.By the law of conservation of energy,$U + K = 0$,where $K$ is the kinetic energy $\frac{1}{2}mv^2$.$-\frac{2Gm(M_1 + M_2)}{d} + \frac{1}{2}mv^2 = 0$$\frac{1}{2}mv^2 = \frac{2Gm(M_1 + M_2)}{d}$$v^2 = \frac{4G(M_1 + M_2)}{d}$$v = \sqrt{\frac{4G(M_1 + M_2)}{d}}$

Class 11 Physics Gravitation Escape Velocity and Escape Energy

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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 a distance $d$ apart. The minimum speed with which a particle of mass $m$ should be projected from a point midway between the two centres so as to escape to infinity is

A
$v = \sqrt {\frac{{4g({M_1} + {M_2})}}{d}} $
B
$v = \sqrt {\frac{{4G({M_1} + {M_2})}}{d}} $
C
$v = \sqrt {4G({M_1 M_2})} $
D
$v = \sqrt {4Gd({M_1} + {M_2})} $

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Escape Velocity and Escape Energy

$A$ ball $A$ of mass $m$ falls to the surface of the earth from infinity. Another ball $B$ of mass $2m$ falls to the earth from a height equal to six times the radius of the earth. The ratio of the velocities of $A$ and $B$ on reaching the earth is:

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The masses and radii of the earth and moon are $(M_1, R_1)$ and $(M_2, R_2)$ respectively. Their centers are at a distance $r$ apart. Find the minimum escape velocity for a particle of mass $m$ to be projected from the midpoint between these two masses.

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If $M$ is the mass of a planet and $R$ is its radius,then in order to become a black hole (where $c$ is the speed of light),which condition must be satisfied?

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$A$ spherical uniform planet is rotating about its axis. The velocity of a point on its equator is $V$. Due to the rotation of the planet about its axis,the acceleration due to gravity $g$ at the equator is $1/2$ of $g$ at the poles. Find the escape velocity of a particle on the planet in terms of $V$.

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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 a distance $d$ apart. The minimum speed with which a particle of mass $m$ should be projected from a point midway between the two centres so as to escape to infinity is

Similar Questions

$A$ ball $A$ of mass $m$ falls to the surface of the earth from infinity. Another ball $B$ of mass $2m$ falls to the earth from a height equal to six times the radius of the earth. The ratio of the velocities of $A$ and $B$ on reaching the earth is:

There are two planets. The ratio of the radii of the two planets is $K$,and the ratio of the acceleration due to gravity of both planets is $g$. What will be the ratio of their escape velocities?

The masses and radii of the earth and moon are $(M_1, R_1)$ and $(M_2, R_2)$ respectively. Their centers are at a distance $r$ apart. Find the minimum escape velocity for a particle of mass $m$ to be projected from the midpoint between these two masses.

If $M$ is the mass of a planet and $R$ is its radius,then in order to become a black hole (where $c$ is the speed of light),which condition must be satisfied?

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