A rocket is fired vertically with a speed of $5 ext{ km/s}$ from the Earth's surface. Approximately

Question

A rocket is fired vertically with a speed of $5	ext{ km/s}$ from the Earth's surface. Approximately how far from the centre of the Earth does the rocket go before returning? (Take $R_E = 6.4 	imes 10^6	ext{ m}$, $g = 9.8	ext{ m/s}^2$)

Professor PC Thomas Classes · Answer

Using conservation of energy: $\frac{1}{2}mv^2 - \frac{GMm}{R_E} = -\frac{GMm}{r_{max}}$, where $r_{max}$ is the maximum distance from the centre. Rearranging gives $r_{max} = \frac{R_E}{1 - v^2/(2gR_E)}$. Substituting $v = 5000	ext{ m/s}$ and evaluating yields $r_{max} \approx 8.0 	imes 10^6	ext{ m}$.