Gravitation
Gravitation covers Newton's inverse-square law, variation of g, Kepler's laws, orbital and escape speeds, and the energy of satellites.
Core rules
- F = Gm₁m₂/r², with G = 6.67×10⁻¹¹ N m² kg⁻².
- g = GM/R² at the surface; at height h, g(h) = g(R/(R+h))²; at depth d, g(d) = g(1 − d/R), and g is zero at the centre.
- Kepler's laws: elliptical orbits; equal areas in equal times; T² ∝ a³.
- Orbital speed v(o) = √(GM/r), about 7.9 km s⁻¹ close to Earth.
- Escape speed v(e) = √(2GM/R) = √(2gR) ≈ 11.2 km s⁻¹, independent of the projectile's mass and launch angle.
- Satellite energy: K = GMm/2r, U = −GMm/r, E = −GMm/2r; bound orbits have negative total energy.
Worked example
For Earth, g = 9.8 m s⁻² and R = 6.4×10⁶ m. Escape speed v(e) = √(2gR) = √(2 × 9.8 × 6.4×10⁶) = √(1.25×10⁸) ≈ 1.12×10⁴ m s⁻¹ = 11.2 km s⁻¹. A satellite in a circular orbit just above the surface would move at v(e)/√2 ≈ 7.9 km s⁻¹.
Traps
Using h alone instead of R + h in the denominator; thinking escape speed depends on mass or launch angle; dropping the negative sign of gravitational potential energy; confusing orbital radius with altitude above the surface.