\begin{aligned}E = KE_{i} + GP E_{i} & = KE_{f} + GP E_{f} \\ \frac{1}{2}mv_{i}^{2} + \frac{−GM_{E}m}{R_{E}} & = \frac{1}{2}mv_{f}^{2} + \frac{−GM_{E}m}{R_{f}} \\ \frac{1}{2}mv_{i}^{2} − \frac{GM_{E}m}{R_{E}} & = \frac{1}{2}m0^{2} − \frac{GM_{E}m}{∞} \\ \frac{1}{2}mv_{i}^{2} − \frac{GM_{E}m}{R_{E}} & = 0 − 0 \\ \frac{1}{2}mv_{i}^{2} − \frac{GM_{E}m}{R_{E}} & = 0 \end{aligned}

\frac{1}{2}mv_{esc}^{2} = \frac{GM_{E}m}{R_{E}}

v_{esc} = \sqrt{\frac{2GM_{E}}{R_{E}}}