Acceleration — Definition, Formula and Worked Examples

High School⏱ 9 min readClassical Mechanics

Acceleration is the rate of change of velocity: a = Δv/Δt. Units, vector direction, sign conventions, and worked examples from cars to free fall.

Quick answer

Acceleration is how quickly velocity changes. A car whose velocity grows by 5 m/s every second has an acceleration of 5 m/s².

$a = \frac{\Delta v}{\Delta t} = \frac{v - u}{t}$

Unit: meters per second squared (m/s²)

Example: A sprinter goes from 2 m/s to 10 m/s in 4 s: a = (10 − 2)/4 = 2 m/s².

What you will learn

Define acceleration as the rate of change of velocity
Use a = Δv/Δt to solve basic problems
Explain why acceleration is a vector quantity
Interpret positive and negative acceleration correctly

What acceleration really measures

Velocity tells you how fast position changes. Acceleration tells you how fast velocity itself changes. Flooring the gas pedal, slamming the brakes, and swerving around a corner are all accelerations — because velocity has both magnitude and direction, changing either counts.

A useful feel for the unit: $9.8\ \mathrm{m/s^2}$ (Earth's gravity) means a falling object's downward velocity grows by $9.8\ \mathrm{m/s}$ every single second.

The formula

Average acceleration over a time interval is the change in velocity divided by the time taken:

a = \frac{\Delta v}{\Delta t} = \frac{v - u}{t}

u = initial velocity, v = final velocity, t = time

Direction matters: acceleration is a vector

Because velocity is a vector, acceleration is one too. Three cases, all acceleration:

Speeding up in your direction of motion → acceleration points forward.
Slowing down → acceleration points backward, opposite the velocity.
Turning at constant speed → acceleration points toward the inside of the turn. The magnitude of velocity never changes, yet you accelerate the whole way around.

That last case surprises everyone: a car cruising around a roundabout at steady 30 km/h is accelerating continuously — its velocity vector keeps rotating.

Velocity vectors at two instants; their difference Δv points inward — that is the acceleration direction.

Reading the sign correctly

Signs are bookkeeping, not physics. If you declare 'forward' positive:

$a = +3\ \mathrm{m/s^2}$ → speeding up forward, or slowing down while moving backward.
$a = -3\ \mathrm{m/s^2}$ → slowing down while moving forward, or speeding up backward.

'Negative acceleration' does not automatically mean 'slowing down' — it means accelerating toward the negative direction. Always sketch your coordinate system first.

Common trap

'Deceleration' is not a technical term — physicists just say acceleration in the opposite direction of motion. Using it loosely causes sign errors.

Constant acceleration and the SUVAT link

When acceleration is constant, velocity changes by equal steps each second, and the average velocity is simply $(u + v)/2$ . That leads straight to the motion equations, the first being:

$v = u + at$

This is just $a = (v-u)/t$ rearranged — the same idea wearing a problem-solving outfit. Every SUVAT equation is built on constant acceleration.

Worked mini-example

A cyclist brakes from $12\ \mathrm{m/s}$ to $4\ \mathrm{m/s}$ in $4\ \mathrm{s}$ . Taking forward as positive:

$a = \frac{4 - 12}{4} = -2\ \mathrm{m/s^2}$

The negative sign says the acceleration opposes the motion — a $2\ \mathrm{m/s^2}$ deceleration. If instead the cyclist had been rolling backward at 12 m/s and sped up backward, the same $-2\ \mathrm{m/s^2}$ would mean speeding up. Context is everything.

Variables and units

Symbol	Name	Unit	Meaning
$a$	Acceleration	m/s²	Rate of change of velocity
$Δv$	Change in velocity	m/s	Final velocity minus initial velocity (v − u)
$Δt$	Time interval	s	Time over which the change happens
$u, v$	Initial / final velocity	m/s	Velocities at the start and end of the interval

Common mistakes

✗ Acceleration means speeding up.

Any change in velocity counts — slowing down and turning are accelerations too. 'Speeding up' is only one of three cases.

✗ Zero velocity means zero acceleration.

A ball at the very top of its toss is instantaneously at rest, yet gravity still accelerates it downward at g. Velocity and acceleration are independent quantities.

✗ Negative acceleration always means slowing down.

It means accelerating in the negative direction. An object moving backward and speeding up has negative acceleration while gaining speed.

Real-world applications

Crash-test engineers measure deceleration in g's to rate how violently a car's velocity changes on impact.
Roller-coaster designers cap lateral acceleration so riders feel thrilled, not injured — typically under 4–5 g.
Rocket guidance computers integrate measured acceleration to track velocity and position when GPS is unavailable.

Frequently asked questions

What is the unit of acceleration?

Meters per second squared (m/s²). Read it as 'meters per second, per second' — how much the velocity (measured in m/s) changes during each second.

Can acceleration be zero while moving?

Yes — cruise control at steady speed on a straight road is zero acceleration with large velocity. Acceleration measures change in velocity, not velocity itself.

What is g in acceleration?

Little-g (g ≈ 9.81 m/s²) is the acceleration due to gravity near Earth's surface. It's also used as a unit: fighter pilots talk about pulling '9 g', meaning nine times Earth's gravitational acceleration.

How is acceleration different from velocity?

Velocity is how fast position changes; acceleration is how fast velocity changes. You feel acceleration (pressed into your seat), never velocity — that's why constant-speed flight feels like standing still.