Constant Acceleration

Displacement — the straight-line distance from start to finish, in a stated direction. A vector, so it can be smaller than the distance travelled, and zero for a round trip.

Velocity — rate of change of displacement. A vector.

Acceleration — rate of change of velocity. A vector, and it does not mean "speeding up": any change in velocity is an acceleration, including slowing down and changing direction at constant speed.

The equations of motion — when acceleration is constant, four equations connect the five quantities s, u, v, a and t.

v = u + at

s = ½(u + v)t

s = ut + ½at²

v² = u² + 2as

Choosing which one — each equation omits exactly one of the five quantities. List what you have and what you want; the equation you need is the one missing the quantity you neither have nor want.

Example: a car accelerates from rest over 50 m and you want its final velocity, knowing the acceleration. You have u, s and a, and you want v — so t is the one you neither have nor want. That points at v² = u² + 2as, with no need to find the time at all.

The condition everything rests on — these equations apply only while the acceleration is constant. Applied to a car that accelerates, cruises and then brakes, they give nonsense unless each phase is treated separately.

Signs and direction — choose a positive direction before you start and stick to it. A car decelerating at 2.0 m s⁻² while travelling in the positive direction has a = −2.0 m s⁻². Dropping the sign is the most common way to get a right-looking wrong answer.

Example: a ball thrown upwards at 5.0 m s⁻¹ with up taken as positive has u = +5.0 m s⁻¹ and a = −9.81 m s⁻². At the top of its flight v = 0, but the acceleration is still −9.81 m s⁻² — gravity does not switch off at the summit.