A particle is fired vertically upward from the earth's surface and it goes up to a maximum height eq

Question

A particle is fired vertically upward from the earth's surface and it goes up to a maximum height equal to the radius of the earth ($R = 6400~km$). Find the initial speed of the particle. (Take $g = 9.8~m~s^{-2}$ at the surface)

Professor PC Thomas Classes · Answer

By conservation of mechanical energy: $\frac{1}{2}mv^2 - \frac{GMm}{R} = 0 - \frac{GMm}{R+h}$. Since maximum height $h = R$, we have $\frac{1}{2}mv^2 = \frac{GMm}{R} - \frac{GMm}{2R} = \frac{GMm}{2R}$. This gives $v = \sqrt{\frac{GM}{R}} = \sqrt{gR}$. Substituting values: $v = \sqrt{9.8 	imes 6400 	imes 10^3} \approx 7.9 	imes 10^3~m~s^{-1} = 7.9~km~s^{-1}$.