Question

An object is projected horizontally with speed $\frac{1}{2} \sqrt{\frac{G M}{R}}$, from a point at height $3 R$ [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

Answer: (C)

At height $3 R$, i.e at distance $4 R$ from the centre of the earth,
$V_{\text {orbital }}=\sqrt{\frac{G M}{r}}$
Here, $r=4 R \Rightarrow V_{0}=\sqrt{\frac{G M}{4 R}}=\frac{1}{2} \sqrt{\frac{G M}{R}}$
Thus, an object taken to a height $3 R$ if projected horizontally with speed $\frac{1}{2} \sqrt{\frac{G M}{R}}$,
will start rotating around earth in a circular orbit.