Question

The ratio of the energy required to raise a satellite upto a height h above the surface of earth to that the kinetic energy of the satellite into the orbit there is (R= radius of earth)

• 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:

$E_1=△U=\frac{mgh}{1+\frac{h}{R}}$ -------------(1)

$E_2$ = energy required to put in orbit

= $\frac{1}{2}m{v}_0^2$ ; where $v_0$ is orbital velocity

= $\frac{1}{2}m\left(\frac{GM}{r}\right)$

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

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

or $E_2=\frac{mgR}{2(1+\frac{h}{R})}$ -----------------(2)

From Eqs . (1) and (2), we have

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