From the story. Homer Hickam went on to work at NASA's Marshall Space Flight Center from 1981 to 1998, where he helped train astronauts. Reaching the Moon or the planets takes more speed than any orbit around Earth. (More in October Sky.)
Throw something up and it slows as it climbs, because gravity pulls it back. But gravity weakens with distance, so the total energy needed to climb out of Earth's “well” is finite. Give an object at least that much kinetic energy and it never returns.
Key idea
Escape condition: $\tfrac12 m v^2 = \dfrac{GMm}{r}$, so $$\role[escape speed]{velocity}{v_\text{esc}} = \sqrt{\frac{2\role[gravitational constant]{constant}{G}\role[Earth's mass]{mass}{M}}{\role[distance from the center]{position}{r}}} = \sqrt2\,\role[orbital speed]{velocity}{v_\text{orbit}}.$$ From Earth's surface it is about $11.2\ \text{km/s}$; from the Moon about $2.4\ \text{km/s}$. Use $r$ from the center.
Worked example: From the Moon
The Moon's mass is $7.342 \times 10^{22}\ \text{kg}$ and its radius $1{,}737\ \text{km}$. Find its escape speed.
- $v = \sqrt{\dfrac{2(6.674 \times 10^{-11})(7.342 \times 10^{22})}{1.737 \times 10^6}} \approx 2{,}380\ \text{m/s}$.
Answer: $\approx 2.38\ \text{km/s}$
Common mistake
Using the altitude above the surface for $r$. Gravity's formulas measure from the center.