Question

The initial velocity $v_i$ required to project a body vertically upward from the surface of the earth to reach a height of $10R,$ where $R$ is the radius of the earth, may be described in terms of escape velocity $v_e$ such that $v_i=\sqrt{\frac{x}{y}}\times v_e.$ The value of $x$ will be

Answer 1

Answer: 10

Here $R=$ radius of the earth
From energy conservation
$\frac{-G{m}_{e}m}{R}+\frac{1}{2}m{v}_i^2=\frac{-G{m}_{e}m}{11R}+0$
$\frac{1}{2}m{v}_i^2=\frac{10}{11}\frac{G{m}_{e}m}{R}$
$V_i=\sqrt{\frac{20}{11}\frac{G{m}_e}{R}}$
$V_i=\sqrt{\frac{10}{11}}{v}_e\{\because$ escape velocity $v_e=\sqrt{\frac{2G{m}_e}{R}}\}$
Then the value of $x=10$