From the story. The boys' exhibit at the 1960 National Science Fair won a gold and a silver medal in propulsion, the science of how rockets push. A push that lasts longer can matter as much as a harder one. (More in October Sky.)
A motor's thrust changes during its burn: often a quick spike, then a steadier push. The total effect is the impulse, the area under the thrust-time curve. Impulse equals change in momentum, so it tells you how much speed the motor can give a rocket of a given mass.
Key idea
$$\role[impulse]{momentum}{J} = \int \role{force}{F}\,\role[tiny time interval]{time}{dt} = \role[average force]{force}{F_\text{avg}}\,\role[burn time]{time}{\Delta t} = \role[change in momentum]{momentum}{\Delta p} = \role{mass}{m}\,\role[change in velocity]{velocity}{\Delta v}.$$ With gravity during a vertical burn: $J_\text{net} = (F_\text{avg} - mg)\,\Delta t$. Commercial model motors are rated by total impulse: class A is up to $2.5\ \text{N}\cdot\text{s}$, and each letter after it allows twice as much.
Worked example: Area under the curve
Thrust rises in a straight line to $12\ \text{N}$ at $0.2\ \text{s}$, drops at once to $4\ \text{N}$, and stays there until $1.2\ \text{s}$. Find the impulse, and the speed it would give a $60\ \text{g}$ rocket with no gravity or drag.
- Triangle: $\tfrac12(0.2)(12) = 1.2$. Rectangle: $4(1.0) = 4.0$. $J = 5.2\ \text{N}\cdot\text{s}$.
- $\Delta v = J/m = 5.2/0.060 \approx 87\ \text{m/s}$.
Answer: $5.2\ \text{N}\cdot\text{s}$, $\approx 87\ \text{m/s}$
Common mistake
Multiplying the peak thrust by the burn time. The spike is brief; only the area counts.
The real Rocket Boys made their own motors, and they had accidents doing it. Don't. Hobby rocketry uses commercially made, certified motors and the National Association of Rocketry's Model Rocket Safety Code.