MOTION IN A LINE
Constant acceleration, drawn three ways: v is the slope of the x–t curve, Δx is the area under v–t. The two kinematic equations and the calculus that ties them together.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A dropped stone, a braking car, a ball thrown straight up — all the same physics: motion along one line under a single, unchanging acceleration. Four quantities, three equations, and everything else follows.
The motion
Start at v₀ and let a act: v = v₀+at and x = v₀t+½at². Negative acceleration slows the object through zero and reverses it.
The numbers
For these controls the stop is t = −v₀/a = 5 s at x = 50 m. The graph window expands to include the full return.
Three graphs
The same motion, drawn three ways. Sweep time and watch: the slope of the x–t curve IS the velocity; the area under the v–t line IS the displacement. Calculus, turned into geometry you can point at.
Audit
Audited: dx/dt reproduces v to a part in ten-million by finite difference; ∫v dt reproduces x by fine Riemann sum; v² = v₀² + 2aΔx holds with residual exactly zero; and the average-velocity shortcut (v₀+v)/2 lands the displacement on the nose.