Question

The ratio of the energy required to raise a satellite upto a height $h$ above the earth to the kinetic energy of the satellite into the orbit is :

• A. $h:R$
• B. $R:2h$
• C. $2h:R$
• D. $R:h$

Answer 1

The correct option is C $2h:R$

Energy required to raise a satellite upto a height $h$,

$E_1=\Delta U=\frac{mgh}{1+\frac{h}{R}}$ ...(i)

$E_2=$ energy required to put in orbit

$=\frac{1}{2}m{v}_0^2$

$=\frac{1}{2}m\left(\frac{GM}{r}\right)$ as $v_0=$ orbital speed

$=\frac{1}{2}m\left(\frac{GM}{R+h}\right)=\frac{1}{2}m\left(\frac{GM}{R^2}\right)\frac{R}{1+\frac{h}{R}}$

$\Rightarrow E_2=\frac{mgR}{2(1+\frac{h}{R})}$ ....(ii)

From equations (i) and (ii), we get

$\frac{E_1}{E_2}=\frac{2h}{R}$