Displacement (Uniform Acceleration)

When acceleration is constant, displacement has two parts: the distance you would cover at your initial velocity alone, plus the extra distance contributed by speeding up, and that extra grows with the square of time. A jet starting its takeoff roll from rest and holding 2 m/s² covers d = 0 + ½(2)(30²) = 900 m in 30 seconds, which is why runways are measured in kilometres.

The ½ appears because the acceleration term is built from the average of a speed that grows linearly from zero. Galileo uncovered the underlying pattern (distances in successive equal time intervals follow the odd numbers 1, 3, 5, 7) by rolling bronze balls down inclined planes. Note that solving for t means solving a quadratic with potentially two positive roots, the first passage and a return, so the calculator gives the earliest positive time and says so; d, v₀ and a are always single-valued.