Question

The escape speed of a body on surface of earth is Ve . If a body is projected from earth with speed Ve?2 ,then at infinity speed of body will be

Answer 1

Dear Student,

If v is the velocity of projected and v' is the velocity at infinity, then we have by energy conservation principle.

$$\begin{gathered} \frac{1}{2}m{v}^2-\frac{GMm}{R}=\frac{1}{2}mv{\text{'}}^2+0 \\ Here,v=2v_e \\ thus. \\ 2{v_e}^2-\frac{GM}{R}=\frac{v{\text{'}}^2}{2} \\ But,v_e=\sqrt{\frac{2GM}{R}}\Rightarrow \frac{{\displaystyle GM}}{{\displaystyle R}}=\frac{{v_e}^2}{2} \\ thus,2{v_e}^2-\frac{{\displaystyle {v_e}^2}}{{\displaystyle 2}}=\frac{{\displaystyle v{\text{'}}^2}}{{\displaystyle 2}} \\ or,v{\text{'}}^2=3{v_e}^2 \\ or,v\text{'}=\sqrt{3}{v}_e\Rightarrow \sqrt{3}\times 11.2km/s=19.4km/s \end{gathered}$$

Regards.