Question
A shell is fired vertically from the earth with speed $$\frac{{{V_{{\text{esc}}}}}}{N},$$ where $$N$$ is some number greater than one and $${{V_{{\text{esc}}}}}$$ is escape speed for the earth. Neglecting the rotation of the earth and air resistance, the maximum altitude attained by the shell will be ($${R_E}$$ is radius of the earth)
A.
$$\frac{{{N^2}{R_E}}}{{{N^2} - 1}}$$
B.
$$\frac{{N{R_E}}}{{{N^2} - 1}}$$
C.
$$\frac{{{R_E}}}{{{N^2} - 1}}$$
D.
$$\frac{{{R_E}}}{{{N^2}}}$$
Answer :
$$\frac{{{R_E}}}{{{N^2} - 1}}$$
Solution :
By conservation of energy
$$\eqalign{
& - \frac{{GMm}}{{{R_E}}} + \frac{1}{2}\frac{m}{{{N^2}}}\frac{{GM}}{{2{R_E}}} = - \frac{{GMm}}{H} \cr
& \Rightarrow H = \frac{{{N^2}{R_E}}}{{{N^2} - 1}} \cr
& {\text{Altitude}} = H - R = \frac{{{R_E}}}{{{N^2} - 1}} \cr} $$