Diagnose numerical overshoot in a stable physical model

The Euler stability analysis in this lesson is optional numerical enrichment. It extends the core Newtonian and resistive-force model by distinguishing a numerical approximation from its exact solution; the detailed stability criterion is not presented as a required AP exam formula.

Separate physics from its numerical approximation

The original ODE is dv/dt=(4−v)/2 with v0=0. Its exact solution 4(1−e^(−t/2)) approaches 4 monotonically from below. Forward Euler replaces each short interval by a tangent step: v[n+1]=v[n]+dt(4−v[n])/2. That approximation is a separate mathematical object with its own error and stability constraints.

Derive the recurrence behavior

Set r=dt/tau, where tau=2 s, and define deviation e[n]=v[n]−4. Subtraction gives e[n+1]=(1−r)e[n], so v[n]=4[1−(1−r)^n]. For 0<r<1, the deviation decays without sign changes. For 1<r<2 it alternates sign while decaying, producing numerical overshoot absent from the exact rest-release solution.

Locate the stability boundary

At r=2 the multiplier is −1, so oscillations never decay. Above 2 its magnitude exceeds one and the numerical sequence diverges. The underlying physical ODE has not become unstable. A negative or enormous Euler result here is evidence of an unsuitable time step, not a newly discovered reversal or a violation of Newton’s law.

Compare at an identical physical time

With r=.5, dt=1 s and four steps reach 4 s. Euler gives 3.75 m/s, while exact 4(1−e⁻²)≈3.45866. Halving dt to.5 s requires eight steps to reach that same 4 s; Euler then gives 4(1−.75⁸)≈3.59955. The smaller error is a convergence check. Holding the step count fixed would compare different physical times and obscure that inference.

Transfer and interpret a special step

At r=.5 the first two values are 2 and 3 m/s. At r=1 Euler reaches equilibrium in one step, but the exact trajectory reaches it only asymptotically. A numerically convenient value is not automatically an accurate time history. Retain the exact reference or a justified refinement study and state which quantity is being compared.

Extend cautiously

This example has a known solution and a linear stability calculation. Nonlinear force models can have changing time scales and event boundaries. A solver must check those limits rather than assume a step size that worked once is universally safe. No physical control system or real-time controller is implemented by this teaching recurrence.

Public numerical transfer practice

Try a new practice case

For v0=0,terminal 4 and Euler ratio r=.5, find velocity after two steps.

Numerical result in the requested units Check this practice result

This public, unsaved check is not a secure assessment. Explain the derivative, integral or boundary in the retained course evidence workspace.

Sources and evidence boundary

Original prose, models and cases, checked 30 October 2026 against the current official AP Physics C Mechanics CED and OpenStax University Physics. Values are supplied or generated, not measured apparatus records. No physical launch, suspension, moving machinery or practical safety validation is requested. Virtual captures and public answers do not certify independent mastery or laboratory participation.