Kepler's Laws & Escape Velocity
- Kepler's 3rd Law: $T^2 \propto a^3$
- Acceleration due to Gravity ($g$):
- At height $h$: $g_h = g\left(1 - \frac{2h}{R}\right)$ ($h \ll R$)
- At depth $d$: $g_d = g\left(1 - \frac{d}{R}\right)$
- Escape Speed ($v_e$):
$$v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR} \approx 11.2\text{ km/s (for Earth)}$$
