The two-body problem. Cosmic velocities
The two-body problem: two bodies moving only under their mutual gravity move about their common centre of mass along a conic section – circle, ellipse, parabola or hyperbola; the shape depends on the total energy: negative gives an ellipse (bound orbit), zero a parabola, positive a hyperbola. The first cosmic velocity v₁ = √(GM/R) is needed to circle near the surface: ≈ 7.9 km/s for the Earth. The second (escape) velocity v₂ = √2 · v₁ ≈ 11.2 km/s is enough to leave the Earth for infinity. The third cosmic velocity v₃ ≈ 16.7 km/s is needed to leave the Solar System from near the Earth (launching in the direction of the Earth’s orbital motion to use its speed). Escape speeds at other bodies: the Moon ≈ 2.4 km/s, Mars ≈ 5.0 km/s, Jupiter ≈ 59.5 km/s.
Paper-and-ball model: roll a ball off a table edge at increasing horizontal speeds and watch it land farther (Newton’s cannonball idea); imagine that at high enough speed it would go round the Earth. Use only a soft ball.