Gravity and orbits formulas

Pull between any two masses. r = center to center. Weight at a height h: m₁ = planet mass (Earth 5.97E24), m₂ = object, r = 6.37E6+h. Touching balls: r = radius 1 + radius 2. Two pulls on one mass: find each F, then add (same way) or subtract (opposite ways). 2× the distance → 1/4 the force. Moon 7.35E22 kg, Sun 1.99E30 kg. Earth to Moon 3.84E8 m, Earth to Sun 1.50E11 m.

Free-fall acceleration at distance r from a planet's center. On the surface: r = planet radius (Earth 6.37E6). At a height h: r = 6.37E6+h. Find the height: solve r, then height = r − 6.37E6. Other planet: clear M, type its mass. Find a planet's mass: clear M (CLEAR), type g and r, solve M. Where 2 pulls cancel (d apart): x = d/(1+√(m₂/m₁)) from m₁. Earth-Moon: 3.46E8 m. Weight = m×g.

Speed of a satellite or moon in a circle orbit. r = planet center to satellite: 6.37E6+height. The satellite's mass does not matter. Around the Sun: M = 1.99E30. Around the Moon: M = 7.35E22. Find a planet's or star's mass from a moon's speed and r: clear M (CLEAR), solve M. Orbit KE = ½mv².

Time for one full orbit (Kepler's 3rd law). T in seconds: hours×3600, days×86400, years×3.156E7. Stays over one spot: T = 24×3600. Height = r − 6.37E6. Oval orbit: r = (closest + farthest)/2. Farthest = 2r − closest. Sun: M = 1.99E30. Find a star's or planet's mass from its moon: clear M (CLEAR), solve M. Two planets, same star, no mass given: clear M, solve M from planet 1, use it for planet 2.

Slowest launch speed that never falls back (no air, no engine). From the surface: R = planet radius (Earth's is filled in). From an orbit or height: R = 6.37E6+height. The object's mass does not matter. Other body: Moon 7.35E22 kg, 1.74E6 m. Sun 1.99E30 kg, 6.96E8 m. Black hole size: type its M, vₑ = 3E8 (light), clear R (CLEAR), solve R.

Gravity potential energy. Always negative; 0 when very far away. r = from the planet center: 6.37E6+height. Work to lift from r₁ to r₂: U₂ − U₁ (solve twice, subtract). Use this, not mgh, when the height is big (like 100 km+). Total energy of an orbit: use E = −GMm / 2r. Launch or falling speeds: use the ½v₁² − GM/r₁ formula.

Total energy (KE + U) of a circle orbit. Always negative. Orbit KE = the same number made positive. Orbit U = 2×E. Energy to move up to a higher orbit: E₂ − E₁ (solve twice, subtract). Energy to put it in orbit from the ground: E − U, with U from U = −GMm / r using r = 6.37E6.

Speed at one distance → speed at another (no air, no engine). Mass cancels. Every r is from the planet center: 6.37E6+height. Shot up, how high: v₂ = 0, solve r₂, then height = r₂ − 6.37E6. Falls from rest: v₁ = 0. Very far away: type 10^30 for that r. Escape speed: use vₑ = √(2GM / R). Comet's closest and farthest points: use v₁r₁ = v₂r₂.

Kepler's 2nd law: closer = faster. Only at an oval orbit's closest point and farthest point. r = distance from the Sun's (or planet's) center. No masses needed. Units cancel: both r in AU, or both speeds in km/s, works too.