KE_{i} + GPE_{i} = KE_{f} + GPE_{f}
\frac{1}{2}mv_{i}^{2} + 0 = 0 + mgH

⇒ H = \frac{v_{i}^{2}}{2g}

KE_{i} + GPE_{i} = KE_{f} + GPE_{f}
\frac{1}{2}mv_{i}^{2} − \frac{GM_{E}m}{R_{i}} = 0 − \frac{GM_{E}m}{R_{f}}

\frac{1}{2}mv_{i}^{2} − \frac{GM_{E}m}{R_{E}} = 0 − \frac{GM_{E}m}{∞}

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

v_{esc} = \sqrt{\frac{2GM}{R}}

R_{BH} = \frac{2GM}{c^{2}}

GPE = − \frac{GMm}{R}