Question

A spaceship is launched into a circular orbit close to earth's surface. The additional velocity that should be imparted to the spaceship in the orbit to overcome the gravitational pull is:
(Radius of earth $=6400km$ and $g=9.8m{s}^{-1}$)

• A. $11.2km{s}^{-1}$
• B. $8km{s}^{-1}$
• C. $3.2km{s}^{-1}$
• D. $1.5km{s}^{-1}$

Answer 1

The correct option is C $3.2km{s}^{-1}$

Given,

Radius of the earth, $R=6400km$

Acceleration due to gravity, $g=9.8m/s^2$

When a satellite is orbitting the earth its velocity is its orbital velocity given by orbital velocity:
$V_0=\sqrt{\frac{GM}{R}}=\sqrt{g}R$ $[\because g=\frac{GM}{R^2}]$

If it has to overcome gravitational pull then its velocity should be escape velocity escape velocity $V_e=\sqrt{\frac{2GM}{r}}=\sqrt{2gR}$

Additional velocity required is $V=V_e-V_0=\sqrt{2gR}-\sqrt{gR}=\sqrt{gR}(\sqrt{2}-1)$

given that $R=64004m=64\times {10}^{6}m$

$g=9.8m/s^2$

So, additional velocity will be:
$V=\sqrt{gR}(\sqrt{2}-1)=\sqrt{9.8\times 6.4\times {10}^6}(\sqrt{2}-1)$

$\Rightarrow V=3.28\times {10}^{3}m/s$
Hence, the correct option is $\left(C\right)$