Question

An object is projected horizontally with speed $\frac{1}{2}\sqrt{\frac{GM}{R}}$, from a point at height $3R$ from surface of earth, where $R$ is radius and $M$ is mass of earth, then object will

• A. fall back on surface of earth by following parabolic path.
• B. fall back on surface of earth by following hyperbolic path.
• C. start rotating around earth in a circular orbit.
• D. escape from gravitational field of earth.

Answer 1

The correct option is C start rotating around earth in a circular orbit.

The orbital velocity of a satellite in circular orbit of radius $r$ of a planet having mass $M$ is given by $$v_{orbital}=\sqrt{\frac{GM}{r}}$$ Orbital velocity at height $3R$, i.e at distance $4R$ from the centre of the earth,

$\Rightarrow v_0=\sqrt{\frac{GM}{4R}}=\frac{1}{2}\sqrt{\frac{GM}{R}}$

Thus, an object taken to a height $3R$, if projected horizontally with speed $\frac{1}{2}\sqrt{\frac{GM}{R}},$ will start rotating around earth in a circular orbit.

Hence, option (c) is correct answer.