Result

Constant-acceleration equations

v = v_{0} + a t, x = x_{0} + v_{0} t + \frac{1}{2} a t^{2}

Statement

Velocity and position as functions of time under constant acceleration, the two forms most problems start from.

Symbols and units

$v_{0}, v$	initial and final velocity	m/s
$x_{0}, x$	initial and final position	m
$a$	acceleration	m/s²
$t$	time	s

When it applies

Constant acceleration only — not valid once a changes with time.

Don't use when

Not valid once acceleration changes during the interval — check the force producing it stays constant throughout.

Derivation

Integrate a = dv/dt over constant a:

v = v_{0} + a t

Integrate v = dx/dt using the result above:

x = x_{0} + v_{0} t + \frac{1}{2} a t^{2}

Eliminating t between the two gives a third form, useful when time is not known:

v^{2} = v_{0}^{2} + 2 a (x - x_{0})

Worked examples

Braking distance

v_0 = 25 m/s, a = −5.0 m/s² (braking), find the stopping distance:

0 = v_{0}^{2} + 2 a Δ x \Rightarrow Δ x = \frac{- v _{0}^{2}}{2 a}

Δ x = \frac{- ( 25 ) ^{2}}{2 ( - 5.0 )} = 62.5 m

Watch out for

Valid only while a stays constant — check that the force producing it (e.g. gravity, or a constant applied force) does not change over the interval considered.