Acceleration

How quickly velocity changes, in size, in direction, or both.

10 minUpdated 21 October 2026average accelerationinstantaneous accelerationdeceleration

Lesson at a glanceShow the key points

Average acceleration = (v₂ − v₁)/(t₂ − t₁); instantaneous acceleration a = dv/dt.
Any change of speed or of direction is an acceleration; uniform circular motion is accelerated.
On a line, v = dx/dt and a = dv/dt = d²x/dt².
If v and a have the same sign the speed increases; opposite signs mean deceleration.
Units: m/s²; dimensions [LT⁻²].

Press the accelerator, hit the brakes, or turn the steering wheel. Each one changes your velocity, and so each one is an acceleration.

When velocity changes

A particle whose velocity stays the same as time passes is in uniform motion. If the velocity changes, the particle is accelerating. Velocity is a vector, so it can change in three ways: in size, in direction, or in both.

If the velocity is $\vec v_1$ at time $t_1$ and $\vec v_2$ at $t_2$ , the average acceleration over that interval is

Average acceleration(3.7)

\vec a_{av} = \frac{\vec v_2 - \vec v_1}{t_2 - t_1}

Like average velocity, it uses only the two end values; how the velocity varied in between does not enter. Shrink the interval to get the instantaneous acceleration, usually just called the acceleration:

Acceleration(3.8)

\vec a = \lim_{\Delta t\to 0}\frac{\Delta\vec v}{\Delta t} = \frac{d\vec v}{dt}

dimension

[LT^{-2}]

, SI unit m/s²

Interactive

Same speed, new direction

A ball moves round a circle at a constant 2 m/s. Its velocity arrow turns, so the change Δv = v₂ − v₁ is not zero. Notice Δv points toward the centre of the circle.

|\vec v_1| = |\vec v_2|

2.00m/s

|\Delta\vec v|

1.37m/s

Angle turned40°

Signs on a straight line

On a straight line, choose it as the x-axis. Position is a single coordinate $x$ , and

v = \frac{dx}{dt} \quad (3.9), \qquad a = \frac{dv}{dt} = \frac{d}{dt}\!\left(\frac{dx}{dt}\right) = \frac{d^2x}{dt^2} \quad (3.10, 3.11)

A positive $v$ means moving toward +x; negative means toward −x. The same goes for $a$ . Whether the particle speeds up or slows down depends on how the two signs compare:

Interactive

Speeding up or slowing down?

Set the starting velocity u and a constant acceleration a, then run. The arrows show v (blue) and a (lavender). Same direction: speeding up. Opposite: slowing down.

t

0.00s

x

0.00m

v

4.00m/s

the car isslowing down

Initial velocity

u

4.0m/s

Acceleration

a

-1.0m/s²

$v$ and $a$ with the same sign → speed increases.
$v$ and $a$ with opposite signs → speed decreases. We call this deceleration.

Deceleration is just acceleration pointing against the motion. An acceleration of 2.0 m/s² toward the east is exactly the same thing as a deceleration of 2.0 m/s² toward the west.

Worked example

A polynomial journey

A particle moves along the x-axis with $x = At^3 + Bt^2 + Ct + D$ , where in SI units A = 2, B = −3, C = 4 and D = 1. Find (a) the dimensions of A, B, C, D, (b) the velocity and (c) the acceleration at t = 3 s, (d) the average velocity and (e) the average acceleration from t = 0 to t = 3 s.

What to remember

1 $\vec a = d\vec v/dt$ ; any change in speed or direction means acceleration.
2On a line: $a = dv/dt = d^2x/dt^2$ .
3Same signs for v and a: speeding up. Opposite signs: slowing down (deceleration).

Check yourself

A ball is thrown straight up. On the way up, its velocity is upward and its acceleration is…

A car moving in the −x direction has a = +3 m/s². It is…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

WatchMotion in a Straight LineCrash Course Physics #1 ReadAccelerationThe Physics Classroom ReadAverage and instantaneous accelerationOpenStax University Physics §3.3 ReadAccelerationHyperPhysics

PreviousVelocity & Position Vectors Next Equations of Motion

Acceleration

How quickly velocity changes, in size, in direction, or both.

10 minUpdated 21 October 2026average accelerationinstantaneous accelerationdeceleration

Lesson at a glanceShow the key points

Average acceleration = (v₂ − v₁)/(t₂ − t₁); instantaneous acceleration a = dv/dt.
Any change of speed or of direction is an acceleration; uniform circular motion is accelerated.
On a line, v = dx/dt and a = dv/dt = d²x/dt².
If v and a have the same sign the speed increases; opposite signs mean deceleration.
Units: m/s²; dimensions [LT⁻²].

Press the accelerator, hit the brakes, or turn the steering wheel. Each one changes your velocity, and so each one is an acceleration.

When velocity changes

If the velocity is $\vec v_1$ at time $t_1$ and $\vec v_2$ at $t_2$ , the average acceleration over that interval is

Average acceleration(3.7)

\vec a_{av} = \frac{\vec v_2 - \vec v_1}{t_2 - t_1}

Acceleration(3.8)

\vec a = \lim_{\Delta t\to 0}\frac{\Delta\vec v}{\Delta t} = \frac{d\vec v}{dt}

dimension

[LT^{-2}]

, SI unit m/s²

Interactive

Same speed, new direction

A ball moves round a circle at a constant 2 m/s. Its velocity arrow turns, so the change Δv = v₂ − v₁ is not zero. Notice Δv points toward the centre of the circle.

|\vec v_1| = |\vec v_2|

2.00m/s

|\Delta\vec v|

1.37m/s

Angle turned40°

Signs on a straight line

On a straight line, choose it as the x-axis. Position is a single coordinate $x$ , and

v = \frac{dx}{dt} \quad (3.9), \qquad a = \frac{dv}{dt} = \frac{d}{dt}\!\left(\frac{dx}{dt}\right) = \frac{d^2x}{dt^2} \quad (3.10, 3.11)

A positive $v$ means moving toward +x; negative means toward −x. The same goes for $a$ . Whether the particle speeds up or slows down depends on how the two signs compare:

Interactive

Speeding up or slowing down?

Set the starting velocity u and a constant acceleration a, then run. The arrows show v (blue) and a (lavender). Same direction: speeding up. Opposite: slowing down.

t

0.00s

x

0.00m

v

4.00m/s

the car isslowing down

Initial velocity

u

4.0m/s

Acceleration

a

-1.0m/s²

$v$ and $a$ with the same sign → speed increases.
$v$ and $a$ with opposite signs → speed decreases. We call this deceleration.

Deceleration is just acceleration pointing against the motion. An acceleration of 2.0 m/s² toward the east is exactly the same thing as a deceleration of 2.0 m/s² toward the west.

Worked example

A polynomial journey

What to remember

1 $\vec a = d\vec v/dt$ ; any change in speed or direction means acceleration.
2On a line: $a = dv/dt = d^2x/dt^2$ .
3Same signs for v and a: speeding up. Opposite signs: slowing down (deceleration).

Check yourself

A ball is thrown straight up. On the way up, its velocity is upward and its acceleration is…

A car moving in the −x direction has a = +3 m/s². It is…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

WatchMotion in a Straight LineCrash Course Physics #1 ReadAccelerationThe Physics Classroom ReadAverage and instantaneous accelerationOpenStax University Physics §3.3 ReadAccelerationHyperPhysics

PreviousVelocity & Position Vectors Next Equations of Motion

When velocity changes#

Signs on a straight line#

Check yourself

Keep exploring

When velocity changes#

Signs on a straight line#

Check yourself

Keep exploring

When velocity changes

Signs on a straight line

When velocity changes

Signs on a straight line