Gravitation

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

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

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

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

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

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

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