Question

When the speed of a satellite near the surface of the earth is increased by $x\%$, it will escape from its orbit, the value of $x$ is:

• A. $11.2$
• B. $41.4$
• C. $27.5$
• D. $34.4$

Answer 1

The correct option is B $41.4$

For a satellite near Earth's surface, the orbital velocity:
$$v_0=\sqrt{\frac{G{M}_e}{R_e}}...\left(1\right)$$ And escape velocity:
$$v_e=\sqrt{\frac{2G{M}_e}{R_e}}...\left(2\right)$$ From equation $\left(1\right)$ and $\left(2\right)$,

$v_e=\sqrt{2}{v}_0$

Satellite will escape from earth's surface if $v_0$ is increased to $v_e$.

So, percentage increase in $v_0$,

$x\%=\frac{v_e-v_0}{v_0}\times 100$

$\Rightarrow x\%=\left(\frac{\sqrt{2}{v}_0-v_0}{v_0}\right)\times 100$

$\Rightarrow x\%=41.4\%$

Hence, option (b) is the correct answer.