Question: 31. The escape velocity from the Earth’s surface is v. The escape velocity from the surface of another planet having a radius, four times that of Earth and same mass density is :
(1) v
(2) 2 v
(3) 3 v
(4) 4 v
correct option is 4
Answer: Option (4)
Explanation:
Escape velocity from the surface of a planet is given by v_e=\sqrt{\frac{2GM}{R}}.
If two planets have the same mass density \rho,
then mass is proportional to volume, so M \propto R^3.
Substituting M \propto R^3 into the escape velocity formula,
we get v_e \propto \sqrt{\frac{R^3}{R}}=\sqrt{R^2}.
Hence, escape velocity is directly proportional to radius, v_e \propto R.
For the given planet, radius is four times that of Earth, so R' = 4R.
Therefore, the escape velocity becomes v'_e = 4v.
Thus, the correct answer is Option (4).