The escape velocity for earth is $v$. A planet having 9 times mass that of earth and radius, 16 times that…
The escape velocity for earth is $v$. A planet having 9 times mass that of earth and radius, 16 times that of earth, has the escape velocity of:
$\frac{v}{3}$
$\frac{2 v}{3}$
$\frac{3 v}{4}$
$\frac{9 v}{4}$
Solution
Escape velopity of object from planet is given by
$\left(v_e\right)_p=\sqrt{\frac{2 G M_p}{R_p}}$
$\therefore \quad\left(v_e\right)_p \propto \sqrt{\frac{M_p}{R_p}}$
Now, $M_p=9 M_e$ and $R_p=16 R_e$ (given)
$\frac{\left(v_e\right)_p}{\left(v_e\right)_e}=\sqrt{\frac{M_p}{R_p} \times \frac{R_e}{M_e}}=\sqrt{\frac{9 M_e}{16 R_e} \times \frac{R_e}{M_e}}=\sqrt{\frac{9}{16}}=\frac{3}{4}$
$\Rightarrow \quad\left(v_e\right)_p=\frac{3}{4} v$