Constant-acceleration motion

Constant acceleration comes from equal velocity changes

Constant acceleration means velocity gains the same vector increment in every equal time interval. It is an approximation, not a universal property of motion; it works when the net force is effectively constant for the interval of interest.

Integrate once, then again

Start with a = dv/dt constant. Rearranging gives dv = a dt; integrating from 0 to t gives v − v₀ = at, hence v(t) = v₀ + at. Since v = dx/dt, integrate dx = (v₀ + at)dt from 0 to t: x − x₀ = v₀t + ½at². The factor ½ appears because velocity grows linearly, so its average over the interval is (v₀ + v)/2 = v₀ + ½at.

A car with v₀ = 5.0 m/s and a = 2.0 m/s² travels in 4.0 s: v = 13 m/s and Δx = 5(4) + ½(2)(16) = 36 m. Units check: m/s·s and m/s²·s² both give metres.

When the formula stops being exact

The equations require one-dimensional motion with acceleration constant in time. If a depends on x or v, integrate the actual function instead. If acceleration varies only mildly, the formula may still be a useful local estimate, but the error grows with the interval.

Work it: A train moving at 18 m/s brakes at −1.5 m/s². Derive its stopping time from v = 0 and its stopping distance from x(t), then check the distance using average velocity.