Escape Velocity: $v_e$ = sqrt$\frac{2GM}{R}$ = sqrt(2gR)

• Independent of mass, direction, and angle of projection
• For Earth: 11.2 km/s
• Derived from energy: \frac{1}{2}$$mv_e^2 = $\frac{GMm}{R}$

Orbital Velocity: $v_o$ = sqrt$\frac{GM}{r}$

• Near surface: $v_o$ = sqrt(gR) ≈ 7.9 km/s
• Decreases with altitude (outer orbits are slower)

Key Relation: $v_e$ = sqrt(2) * $v_o$ (at any altitude)

• To escape from circular orbit, increase speed by factor sqrt(2)
• Additional speed needed: $v_o$(sqrt(2) - 1) ≈ 0.414*$v_o$

For a body projected at speed v (v < $v_e$): Maximum height h = $v^2$*$\frac{R}{2gR - v^2}$ At v = $v_e$: h -> infinity At v = $v_e$/2: h = R/3