The kinematic equations

SUVAT equationsequations of motionbig five kinematics

The five equations of motion under constant acceleration, and the trick for choosing between them: pick the one missing the quantity you neither have nor want.

Which one do I use? Pick the equation missing what you neither have nor want

Equation	Leaves out
$v = v_0 + a t$	Displacement
$d = v_0 t + \tfrac{1}{2} a t^2$	Final velocity
$v^2 = v_0^2 + 2 a d$	Time
$d = \frac{v_0 + v}{2} \, t$	Acceleration
$d = v t - \tfrac{1}{2} a t^2$	Initial velocity

Final Velocity (Uniform Acceleration)

v = v_0 + a t

Final velocity after accelerating uniformly from an initial velocity for a given time.

Displacement (Uniform Acceleration)

d = v_0 t + \tfrac{1}{2} a t^2

Distance travelled under constant acceleration, starting from an initial velocity, over a time t.

Velocity-Displacement Relation (v² = v₀² + 2ad)

v^2 = v_0^2 + 2 a d

Links initial and final speeds to acceleration and displacement without involving time.

Displacement from Average Velocity

d = \frac{v_0 + v}{2} \, t

Displacement as the average of initial and final velocities multiplied by the elapsed time, valid for uniform acceleration.

Displacement from Final Velocity (d = vt − ½at²)

d = v t - \tfrac{1}{2} a t^2

The fifth kinematic equation: displacement from the FINAL velocity and the time, for when the starting speed is the unknown.

How they fit together

These five describe every motion with constant acceleration, and they are not five independent facts — they are one fact written five ways. Start from a = Δv/Δt and the definition of average velocity, and the rest follow by substitution. That is why they share the same five quantities: initial velocity, final velocity, acceleration, displacement and time.

The whole skill is choosing. Each equation leaves exactly one of the five out, so list what you have, note what you want, and the quantity that is neither is the one to omit — that names your equation. A problem giving you speeds and a distance but no clock is asking for v² = v₀² + 2ad, the one without time. They only apply while acceleration is constant: the moment a rocket burns fuel or a parachute opens, the motion has to be split into segments with these applied to each.