When acceleration remains constant, motion in one dimension has a particularly simple structure: velocity changes linearly with time and position changes quadratically.

Let the initial position and velocity at $t=0$ be $x_0$ and $v_0$, and let the constant acceleration be $a$.

Velocity

Because velocity changes by the same amount during equal time intervals,

$$v=v_0+at.$$

Position

The displacement is

$$x-x_0=v_0t+\frac12at^2,$$

so

$$x=x_0+v_0t+\frac12at^2.$$

Eliminating time

Combining the constant-acceleration relations also gives

$$v^2=v_0^2+2a(x-x_0).$$

This form is useful when time is not one of the known quantities.

Graphs

For constant acceleration, the acceleration-time graph is horizontal. The velocity-time graph is a straight line whose slope is $a$, and the displacement over a time interval is the signed area under the velocity-time graph.

The position-time graph is parabolic when $a\ne0$.

The signs of $x$, $v$ and $a$ must all refer to the same chosen positive direction; the equations do not require motion itself to remain in that direction.