Question

An artificial satellite is moving around earth in a circular orbit with speed equal to one fourth the escape speed of a body from the surface of earth. The height of satellite above earth is : ($R$ is radius of earth)

• A. $3R$
• B. $5R$
• C. $7R$
• D. $8R$

Answer 1

The correct option is D $8R$

Let's consider,

$r=$ height of the satellite above the earth

$R=$ radius of the earth

$G=$ gravitational constant

$g=$ acceleration due to gravity

The escape velocity of a body from the surface of earth,

$v_e=\sqrt{\frac{2GM}{R}}$

$\frac{1}{4}.v_e=\frac{1}{4}\sqrt{\frac{2GM}{R}}$. . . . . .(1)

The orbital velocity of the satellite is given by

$v_o=\sqrt{\frac{GM}{r}}$. . . . . .(2)

Equation equation (1) and (2), we get

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

$\frac{GM}{r}=\frac{1}{16}.\frac{2GM}{R}$

$r=8R$

The correct option is D.