Physics by Lamhi: Not Your Boring Physics

1. Kepler's Three Laws

First law (orbits): every planet moves in an ellipse, Sun at one focus.
Second law (areas): the Sun–planet line sweeps equal areas in equal times — angular momentum about the Sun stays constant.
Third law (periods): $T^2 \propto a^3$ , same constant for every planet around the Sun.

Universal law of gravitation

F = \dfrac{Gm_1m_2}{r^2}

$G = 6.674\times10^{-11}\ \text{N m}^2\text{kg}^{-2}$ — always attractive, always inverse-square.

Reproduces Kepler's third law

T^2 = \left(\dfrac{4\pi^2}{GM}\right)r^3

g at the surface

g = \dfrac{GM}{R^2}

Height h (h≪R): $g(h) \approx g\left(1-\dfrac{2h}{R}\right)$ — falls off twice as fast per metre as depth.
Depth d: $g(d) = g\left(1-\dfrac{d}{R}\right)$ — a straight line, zero at the centre.

Gravitational P.E.

U(r) = -\dfrac{GMm}{r}

Escape speed

v_e = \sqrt{\dfrac{2GM}{R}}

Independent of launched mass $m$ and of launch direction — only launch speed matters.

Orbital velocity

v_{\text{orbital}} = \sqrt{\dfrac{GM}{r}}

K = \dfrac{GMm}{2r} \qquad U = -\dfrac{GMm}{r} \qquad E = K+U = -\dfrac{GMm}{2r} = -K = \dfrac{U}{2}

Every bound circular orbit: total energy negative, equals $-K$ , equals $U/2$ — for any $r$ , $m$ , or $M$ .

v_e = \sqrt{2}\times v_{\text{orbital}}

Launch speed = $v_{\text{orbital}}$ : circular orbit.
Between $v_{\text{orbital}}$ and $v_e$ : elliptical (bound) orbit.
At or above $v_e$ (= $\sqrt{2}\,v_{\text{orbital}}$ ): orbit opens up — a one-way escape, not a closed path.

Geostationary radius condition

r = \left(\dfrac{GMT^2}{4\pi^2}\right)^{1/3}

Geostationary: equatorial plane + west-to-east + $T=24\text{ h}$ → one unique $r$ (altitude ≈ 35,900 km).
Polar: low altitude, period ≈ 100 min, passes near both poles — images the whole planet over many orbits.
Weightlessness: satellite and contents in free fall together — normal force vanishes, scale reads zero.

Full derivations and worked examples: detailed notes →