Gravitation — Advanced Multi-Step Practice (Hard)

Quick Recap — Advanced Gravitation (Multi-step)

• Escape / orbital numbers: $v_e=\sqrt{2gR}$, $v_o=\sqrt{gR}$ (near surface); with $g=10$, $R=6.4\times10^6$ m these give $\approx11.3$ km/s and $\approx8$ km/s.
• Scaling: $g\propto\rho R$; $v_e\propto R\sqrt{\rho}$; $T^2\propto r^3$ (Kepler).
• Energy to raise a satellite from surface to orbit $r$: $\Delta E=\left(-\dfrac{GMm}{2r}\right)-\left(-\dfrac{GMm}{R}\right)$.
• Speed to escape from an orbit (not the surface): $v=\sqrt2\times$(local orbital speed).

Worked mini-example. For $g=10$, $R=6.4\times10^6$ m: $v_e=\sqrt{2gR}=\sqrt{1.28\times10^8}\approx1.13\times10^4$ m/s $=11.3$ km/s.