A particle is fired vertically upward with a speed of $15~km~s^{-1}$. With what approximate speed wi

Question

A particle is fired vertically upward with a speed of $15~km~s^{-1}$. With what approximate speed will it move in interstellar space? Assume only the earth's gravitational field. (Escape speed from earth is $11.2~km~s^{-1}$)

Professor PC Thomas Classes · Answer

By conservation of mechanical energy: $\frac{1}{2}mv^2 - \frac{GMm}{R} = \frac{1}{2}mv'^2$. Since escape velocity $v_e^2 = \frac{2GM}{R}$, we have $v'^2 = v^2 - v_e^2 = 15^2 - 11.2^2 = 225 - 125.44 \approx 99.56$. Therefore, $v' \approx 10.0~km~s^{-1}$.