Numerical Kinematics

Start from a force balance and initial conditions

For a constant mass, use net force divided by mass to obtain acceleration. Velocity comes from integrating acceleration; position comes from integrating velocity. Initial velocity and initial position are required in addition to the samples.

Forward Euler uses the old state

For time step Δt, explicit Euler uses acceleration and velocity at the start of each interval. Update position from the old velocity if you intend this method; using the newly updated velocity is a different scheme.

Trapezoids use both ends of an interval

For sampled acceleration, average adjacent values before multiplying by the interval width. Integrating sampled velocity similarly estimates position. For smooth, accurately sampled curves, composite trapezoidal error is second order.

Higher order still needs good data

Composite Simpson's rule uses pairs of equally spaced intervals and requires an even number of intervals in its standard form. Its fourth-order accuracy assumes sufficient smoothness. Smaller steps do not remove sensor bias, and numerical integration can accumulate measurement errors.

Worked example

A 2 kg mass experiences net force F(t) = 4t N for 0 ≤ t ≤ 1 s, with v₀ = 1 m/s and x₀ = 0. Use two 0.5 s steps to estimate final velocity.

1. 1.Compute acceleration at the sample times.

   $$\begin{array}{c}a(t)=2t\\ a(0)=0\\ a(0.5)=1\\ a(1)=2[m/{\mathrm{s}}^2]\end{array}$$

2. 2.Euler uses the left endpoint on each step.

   $$v_{\mathrm{Euler}}(1)=1+0(0.5)+1(0.5)=1.50m/s$$

3. 3.Trapezoids use both endpoints and are exact for this linear acceleration.

   $$v_{\mathrm{trap}}(1)=1+\frac{0+1}{2}(0.5)+\frac{1+2}{2}(0.5)=2.00m/s$$

The exact velocity is 2.00 m/s. With this coarse step, Euler underestimates it by 0.50 m/s; trapezoidal integration is exact for this particular acceleration.