Question

A spaceship is launched into a circular orbit of radius $R$ close to the surface of earth. The additional velocity to be imparted to the spaceship in the orbit to overcome the earth's gravitational pull is
(Take $\sqrt{2}=1.414$ and $g=$ acceleraton due to gravity )

• A. $1.414Rg$
• B. $1.414\sqrt{Rg}$
• C. $0.414Rg$
• D. $0.414\sqrt{gR}$

Answer 1

The correct option is D $0.414\sqrt{gR}$

Since the spaceship is in an orbit of radius $R$, its orbital speed will be

$v_o=\sqrt{\frac{GM}{R}}$

The velocity required to overcome the earth's gravitational pull is the escape velocity. So, escape velocity will be

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

Additional velocity required,

$\Delta v=v_e-v_o=\sqrt{\frac{2GM}{R}}-\sqrt{\frac{GM}{R}}$

Since, $g=\frac{GM}{R^2}$

$\Rightarrow \Delta v=(\sqrt{2}-1)\sqrt{\frac{GM}{R}}=0.414\sqrt{gR}$

Hence, option $\left(d\right)$ is correct.