(C) The orbital velocity of a satellite at a height $h$ above the Earth's surface is given by $V_{0} = \sqrt{\frac{GM}{R+h}}$.The escape velocity from

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(C) The orbital velocity of a satellite at a height $h$ above the Earth's surface is given by $V_{0} = \sqrt{\frac{GM}{R+h}}$.The escape velocity from the Earth's surface is given by $V_{e} = \sqrt{\frac{2GM}{R}}$.According to the problem,the orbital speed is half the escape velocity: $V_{0} = \frac{V_{e}}{2}$.Substituting the expressions: $\sqrt{\frac{GM}{R+h}} = \frac{1}{2} \sqrt{\frac{2GM}{R}}$.Squaring both sides: $\frac{GM}{R+h} = \frac{1}{4} \times \frac{2GM}{R}$.Simplifying the equation: $\frac{1}{R+h} = \frac{1}{2R}$.Therefore,$R+h = 2R$,which gives $h = R$.

Class 11 Physics Gravitation Motion of Satellites in Circular Orbits and Planets in Elliptical Orbits

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If an artificial satellite is moving in a circular orbit around the earth with a speed equal to half the magnitude of the escape velocity from the earth,the height of the satellite above the surface of the earth is

A
$2R$
B
$\frac{R}{2}$
C
$R$
D
$\frac{R}{4}$

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Motion of Satellites in Circular Orbits and Planets in Elliptical Orbits

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$A$ particle moves in a circular orbit of radius $r$ under a central attractive force $F = -\frac{k}{r}$,where $k$ is a constant. The periodic time of this motion is proportional to:

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$A$ planet of mass $m$ is moving around a star of mass $M$ and radius $R$ in a circular orbit of radius $r$. The star abruptly shrinks to half its radius without any loss of mass. What change will be there in the orbit of the planet?

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$A$ satellite of mass $m$ is revolving close to the surface of a planet of density $d$ with time period $T$. The value of the universal gravitational constant $G$ in terms of $d$ and $T$ is given by:

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$A$ body moves in a circular orbit of radius $R$ under the action of a central force. The potential due to the central force is given by $V(r) = kr$ ($k$ is a positive constant). The period of revolution of the body is proportional to:

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If an artificial satellite is moving in a circular orbit around the earth with a speed equal to half the magnitude of the escape velocity from the earth,the height of the satellite above the surface of the earth is

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Two stars of masses $M$ and $2M$ are at a distance $d$ apart and are revolving around their common center of mass. The angular velocity of the system of the two stars is ($G$ is the universal gravitational constant).

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