Complete Formula Reference

Newton's Universal Law

$F = \frac{G m_1 m_2}{r^2} \quad [G] = [\text{M}^{-1}\ \text{L}^3\ \text{T}^{-2}] \quad \text{N m}^2\ \text{kg}^{-2}$

Acceleration Due to Gravity at Surface

$g = \frac{GM}{R^2} \quad [\text{M}^0\ \text{L}^1\ \text{T}^{-2}] \quad \text{m s}^{-2}$

Key Substitution: $GM = gR^2$ — eliminates G and M from all satellite problems.

g at Altitude h (exact)

$g'_h = \frac{gR^2}{(R+h)^2} \quad [\text{M}^0\ \text{L}^1\ \text{T}^{-2}] \quad \text{m s}^{-2}$

g at Altitude h (approximate, $h \ll R$)

$g'_h \approx g\!\left(1 - \frac{2h}{R}\right) \quad \text{valid only for}\ h \ll R$

g at Depth d

$g'_d = g\!\left(1 - \frac{d}{R}\right) \quad [\text{M}^0\ \text{L}^1\ \text{T}^{-2}] \quad \text{m s}^{-2}$

g at Latitude λ

$g_{\text{eff}} = g - R\omega^2\cos^2\!\lambda \quad [\text{M}^0\ \text{L}^1\ \text{T}^{-2}] \quad \text{m s}^{-2}$

Kepler's Third Law

$T^2 = \frac{4\pi^2}{GM}\,r^3 \quad [\text{M}^0\ \text{L}^0\ \text{T}^2] \quad \text{s}^2$

Gravitational Potential Energy

$U = -\frac{GMm}{r} \quad [\text{M}^1\ \text{L}^2\ \text{T}^{-2}] \quad \text{J}$

Gravitational Potential

$V = -\frac{GM}{r} \quad [\text{M}^0\ \text{L}^2\ \text{T}^{-2}] \quad \text{J kg}^{-1}$

Escape Velocity

$v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR} \quad [\text{M}^0\ \text{L}^1\ \text{T}^{-1}] \quad \text{m s}^{-1}$

Earth: $v_e \approx 11{,}200\ \text{m s}^{-1} = 11.2\ \text{km s}^{-1}$

Orbital Velocity

$v_0 = \sqrt{\frac{GM}{r}} = \sqrt{\frac{gR^2}{r}} \quad [\text{M}^0\ \text{L}^1\ \text{T}^{-1}] \quad \text{m s}^{-1}$

Near surface: $v_0 = \sqrt{gR} \approx 7{,}920\ \text{m s}^{-1} = 7.9\ \text{km s}^{-1}$

Relation Between Escape and Orbital Velocity

$v_e = \sqrt{2}\,v_0 \quad \Rightarrow \quad \frac{v_e}{v_0} = \sqrt{2} \approx 1.414$

Orbital Period

$T = \frac{2\pi r}{v_0} = 2\pi\sqrt{\frac{r^3}{GM}} \quad [\text{M}^0\ \text{L}^0\ \text{T}^1] \quad \text{s}$

Near surface: $T = 2\pi\sqrt{R/g} \approx 5{,}073\ \text{s} \approx 84.6\ \text{min}$

Satellite Kinetic Energy

$KE = \frac{GMm}{2r} = \frac{mv_0^2}{2} \quad [\text{M}^1\ \text{L}^2\ \text{T}^{-2}] \quad \text{J}$

Satellite Potential Energy

$PE = -\frac{GMm}{r} = -2KE \quad [\text{M}^1\ \text{L}^2\ \text{T}^{-2}] \quad \text{J}$

Satellite Total Energy

$E = -\frac{GMm}{2r} = -KE \quad [\text{M}^1\ \text{L}^2\ \text{T}^{-2}] \quad \text{J}$

Dimensional Analysis: Deriving G's Dimensions

From $F = Gm_1 m_2 / r^2$:

$[G] = \frac{[F][r^2]}{[m_1][m_2]} = \frac{\text{kg m s}^{-2} \cdot \text{m}^2}{\text{kg} \cdot \text{kg}} = \text{kg}^{-1}\ \text{m}^3\ \text{s}^{-2} = [\text{M}^{-1}\ \text{L}^3\ \text{T}^{-2}]$

SI Unit Consistency Check for $v_e$

$v_e = \sqrt{2gR}\ \Rightarrow\ [v_e] = \sqrt{\text{m s}^{-2} \cdot \text{m}} = \sqrt{\text{m}^2\ \text{s}^{-2}} = \text{m s}^{-1}\ \checkmark$