Rating a Model Rocket Motor

Model rocket motors are graded by total impulse, the area under the thrust curve. A test stand only samples the thrust every so often, so the area is estimated from samples: trapezoids join them with straight lines, and Simpson's rule fits parabolas, which is far more accurate.

How it works

Approximating definite integrals

In one sentence

With $n$ (even) subintervals of width $\Delta x$ , Simpson's rule is $S_n = \frac{\Delta x}{3}\left[f(x_0) + 4f(x_1) + 2f(x_2) + \cdots + 4f(x_{n-1}) + f(x_n)\right]$ , with error at most $\frac{K(b-a)^5}{180n^4}$ where $|f^{(4)}| \le K$ .

General formula

{\htmlData{sym=0}{I}} = \int_0^{\htmlData{sym=2}{{T}}} {\htmlData{sym=1}{F(t)}}\,dt \approx \frac{{\htmlData{sym=3}{\Delta t}}}{3}\left[F_0 + 4F_1 + 2F_2 + \cdots + 4F_{{\htmlData{sym=4}{n}}-1} + F_{\htmlData{sym=4}{n}}\right]

Straight segments miss the curvature; parabolas through each three points follow it, which is why Simpson's rule converges so much faster.

What the symbols mean

$I$
Total impulse
Newton-seconds of push: the motor's rating.
$F(t)$
Thrust
Force in newtons at time t.
$T$
Burn time
How long the motor fires.
$\Delta t$
Sample spacing
Seconds between thrust readings.
$n$
Samples
Number of intervals (even, for Simpson's rule).

The test stand records thrust $F$ every $\Delta t = T/n$ seconds.
Trapezoid rule: join neighboring samples with straight lines and add the trapezoids, $\frac{\Delta t}{2}\left[F_0 + 2F_1 + \cdots + 2F_{n-1} + F_n\right]$ .
Simpson's rule: fit a parabola through each three samples, which gives the weights $1, 4, 2, 4, \ldots, 4, 1$ .
Simpson's error shrinks like $\Delta t^4$ , the trapezoids' like $\Delta t^2$ : halve the spacing and Simpson's gets 16 times better.

Learn the math

Approximating Definite IntegralsCalc II · Lesson 10, Integration Techniques