Question

A satellite of moon revolves around it in a radius n times the radius of moon (R). Due to cosmic dust it experiences a resistance $f=a{v}^2$. Find how long it will stay in the orbit.

• A. $\frac{m}{\alpha \sqrt{\frac{GM}{R}}}\sqrt{n}$
• B. $\frac{m}{\alpha }\sqrt{\frac{R}{GM}}(\sqrt{n}-1)$
• C. $\frac{m}{\alpha }\frac{(\sqrt{n}-1)}{v}$
• D. $\frac{m}{\alpha }\frac{v_l}{v_f^2}$

Answer 1

Answer: (B)

$\frac{m}{\alpha }\sqrt{\frac{R}{GM}}(\sqrt{n}-1)$

$m\frac{dv}{dt}=a{v}^{2}dtor\frac{dv}{v^2}=\frac{\alpha dt}{m}$
and $v=\sqrt{\frac{GM}{r}}$.
We know $v_i=\sqrt{\frac{GM}{nR}}$ and $v_f=\sqrt{\frac{GM}{R}}$
$\therefore {\int }_{v_i}^{v_f}\frac{dV}{V^2}=\int \frac{\alpha dt}{m}$
or $t=\frac{m}{\alpha }[\frac{1}{v_i}-\frac{1}{v_f}]=\frac{m}{\alpha \sqrt{\frac{GM}{R}}}[\sqrt{n}-1]$