v = u + at — Velocity from Acceleration and Time

The first SUVAT equation: final velocity from initial velocity, acceleration and time — with a worked example.

v = u + at

Final velocity equals initial velocity plus acceleration times time.

What it means

Final velocity equals initial velocity plus the velocity gained from accelerating. Each second of acceleration $a$ adds $a$ metres per second to the speed — so velocity grows linearly with time under constant acceleration.

Variables

Symbol	Name	Unit	Meaning
$v$	Final velocity	m/s	Final velocity
$u$	Initial velocity	m/s	Velocity at time zero
$a$	Acceleration	m/s²	Rate of change of velocity
$t$	Time	s	Duration of the acceleration

Assumptions

Acceleration is constant over the time interval
Motion is along a straight line
u, v and a share the same sign convention

Derivation

Acceleration is defined as the rate of change of velocity, $a = \Delta v/\Delta t$ . Rearranging $\Delta v = a\Delta t$ and writing $\Delta v = v - u$ gives $v = u + at$ directly.

Worked example

Problem

A car starts from rest and accelerates at a constant 3 m/s². What is its velocity after 8 seconds?

Given:

u = 0 m/s (starts from rest)
a = 3 m/s²
t = 8 s

Write the equation: $v = u + at$
Substitute: $v = 0 + 3 \times 8$
Calculate: $v = 24\ \mathrm{m/s}$

24 m/s

After 8 seconds of steady acceleration the car is moving at 24 m/s — about 86 km/h. The velocity grows linearly because each second adds 3 m/s.