MODULE 05 / CALCULUS & ENGINEERING TIER
Forces That Change
When force depends on speed or position, motion is the solution of a differential equation.
Learning objective
Conditions · locked for this activity
The experiment
ReadyEnergy · total at release J
■ Kinetic J■ Spring J■ Removed by damper J
Motion graphs
LIVEAxes are fixed to the exact solution for the whole run.
Show the math
Follows the run while it plays; drag to inspect any time already observed.
| Slope of x(t) at t* | m/s | = v(t*) m/s |
|---|---|---|
| Slope of v(t) at t* | m/s² | = a(t*) m/s² |
| Area under v(t), 0 → t* | m | = Δx m |
| Area under a(t), 0 → t* | m/s | = Δv m/s |
Slopes are measured from the curves by finite differences and areas by the trapezoid rule over the recorded samples; the right-hand column is the exact solution. Tangents appear on the x and v graphs; shaded areas on the v and a graphs.
Numerical methods
vs EXACTError = |x_numerical − x_exact|. Halve h and watch how far each curve drops: about 2× for Euler, 16× for RK4.
Exact samples every 0.01 s for the whole run, plus the selected methods at their own steps. Open in MATLAB with readtable or in Python with pandas.read_csv.
Results
Reflect on the evidence
Instructor-authored model answer
One-dimensional motion, SI units. Drag scenario: F ≥ 0 and v₀ ≥ 0; observation ends at 10 s or x = 50 m. Spring scenario: ideal massless spring and linear damper on a frictionless floor; observation 20 s. All curves are evaluated from exact solutions; numerical methods are shown only for comparison.