Question

The minimum speed required by a rocket to put a satellite in space is :

• A. $11.2km/s$
• B. $11.2m/s$
• C. $11.2miles/h$
• D. $11.2km/min$

Answer 1

The correct option is A

$11.2km/s$

Step - 1 : Consider

Radius of earth, $R=6371km=6.371\times {10}^{6}m$

Gravitational constant, $G=6.6743\times {10}^{-11}{m}^{3}k{g}^{-1}{s}^{-2}$

Mass of the earth, $M=5.972\times {10}^{24}kg$

Step - 2 :Formula

The total energy of an object in orbit =$\frac{-GMm}{2r}$

It has the least energy if $r=R$ i.e. earth's surface.

The total energy of object on earth's surface = $\frac{-GMm}{R}$

So, the remaining energy is to be provided by kinetic energy = $\frac{m{v}^2}{2}$

$\Rightarrow \frac{-GMm}{2r}-\frac{-GMm}{r}=\frac{m{v}^2}{2}$

$\Rightarrow v=\sqrt{\frac{GM}{R}}$

Step - 3 : The minimum speed required by a rocket to put a satellite in space

$$\begin{gathered} \Rightarrow v=\sqrt{\frac{GM}{R}} \\ \Rightarrow v=\sqrt{\frac{6.67\times {10}^{-11}\times 5.972\times {10}^{24}}{6.371\times {10}^6}} \\ \Rightarrow v=11200m/s=11.2km/s \end{gathered}$$

Hence option A is correct.