Level 7: Energy and orbits

Gravity and orbits · level 7 of 8 · rough age 15-16 (a guide for parents; anyone can start anywhere)

This level looks at orbits through energy. You find the energy gravity stores at any distance, derive escape speed, see why Kepler's orbits are ellipses and how speed changes along them, use the vis-viva equation to get a speed anywhere on an orbit, and put it all together in the half-ellipse that carries a ship from Earth to Mars in 259 days, and the launch windows that come every 26 months.

Lesson	You will be able to
Gravitational energy	Use \(U = -GMm/r\), find the energy to lift a kilogram off Mars, and the energy of an orbit
Escape speed	Derive \(v_\text{esc} = \sqrt{2GM/R}\) and find it from a surface or an orbit
Ellipses	Describe an elliptical orbit by \(a\) and \(e\), use Kepler's first and second laws, and the orbit of Mars
Vis-viva	Derive and use \(v^2 = GM(2/r - 1/a)\) for speeds anywhere on an orbit
Hohmann transfer	Work out the transfer from Earth to Mars: its shape, its 259 days and the speed changes it needs
Launch windows	Find the phase angle for launch, the 780-day synodic period, and the wait at Mars before coming home

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