Key Equations

Key Equations

Newton’s law of gravitation	${\overrightarrow{F}}_{12}=G\frac{m_1{m}_2}{r^2}{\widehat{r}}_{12}$
Acceleration due to gravity at the surface of Earth	$g=G\frac{M_{\text{E}}}{r_{}^2}$
Gravitational potential energy beyond Earth	$U=-\frac{G{M}_{\text{E}}m}{r}$
Conservation of energy	$\frac{1}{2}m{v}_1^2-\frac{GMm}{r_1}=\frac{1}{2}m{v}_2^2-\frac{GMm}{r_2}$
Escape velocity	$v_{\text{esc}}=\sqrt{\frac{2GM}{R}}$
Orbital speed	$v_{\text{orbit}}=\sqrt{\frac{{\text{GM}}_{\text{E}}}{r}}$
Orbital period	${\rm T}=2\pi \sqrt{\frac{r^3}{{\text{GM}}_{\text{E}}}}$
Energy in circular orbit	$E=K+U=-\frac{Gm{\text{M}}_{\text{E}}}{2r}$
Conic sections	$\frac{\alpha }{r}=1+e\text{cos}\theta$
Kepler’s third law	${{\rm T}}^2=\frac{4{\pi }^2}{GM}{a}^3$
Schwarzschild radius	$R_{\text{S}}=\frac{2GM}{c^2}$