Science

What is acceleration, and what is the formula?

Quick answer

Acceleration is how quickly velocity changes. a = (v − u) ÷ t, the change in velocity divided by the time taken, measured in metres per second squared (m/s²).

Ages 13–16Secondary

In one sentence

Acceleration is how fast something's velocity changes: speeding up, slowing down or changing direction.

Explained

When a bus pulls away from a stop, you feel pushed back into your seat. When it brakes, you lurch forwards. Both times the bus is accelerating: its velocity is changing. In everyday language "accelerate" means speed up, but in physics it means any change in velocity, so slowing down and turning a corner count too.

Acceleration measures how much the velocity changes every second. The formula is:

a = (v − u) ÷ t

u is the starting (initial) velocity in m/s
v is the final velocity in m/s
t is the time taken in seconds (s)
a is the acceleration in metres per second squared (m/s²)

In the US the same formula is often written a = (v_f − v_i) ÷ t or a = Δv ÷ Δt, where Δ means "change in".

A car pulling away, a rocket launching and a plane on the runway are all accelerating.

The unit m/s² looks strange, but it reads naturally: an acceleration of 3 m/s² means the velocity goes up by 3 m/s every second. A car starting from rest with that acceleration is doing 3 m/s after 1 s, 6 m/s after 2 s, 9 m/s after 3 s, and so on. Rearranged, this gives the very useful v = u + at.

If the velocity goes down, v − u is negative, so the acceleration is negative. This is often called deceleration: a bike slowing from 8 m/s to 0 in 4 s has an acceleration of −2 m/s², or a deceleration of 2 m/s².

Acceleration is a vector, because velocity has a direction (see speed, distance and time). That is why a car going round a bend at a steady speed is still accelerating: its direction keeps changing. For something moving in a circle, this acceleration points to the centre and is called centripetal acceleration, a = v² ÷ r.

What causes acceleration? A resultant force. Newton's second law says F = m × a, so a = F ÷ m: a bigger force gives a bigger acceleration, and a bigger mass gives a smaller one. Pushing an empty shopping trolley is easy; a full one accelerates far less for the same push. More on this in forces.

Near the Earth's surface, anything falling freely (with no air resistance) accelerates at about 9.8 m/s². This is the acceleration due to gravity, g (often rounded to 10 m/s² in school questions). On the Moon it is only about 1.6 m/s². See gravity.

Two more equations help when you do not know the time. For constant acceleration: v² = u² + 2as, where s is the distance in metres; and, starting from rest, s = ½at². These belong to the "suvat" equations used in GCSE and A-level physics.

On a velocity–time graph, the gradient (slope) of the line is the acceleration, and the area under the line is the distance travelled. A straight sloping line means constant acceleration; a flat line means constant velocity (zero acceleration).

Examples

Situation	Start velocity u	Final velocity v	Time t	Acceleration a
Car pulling away	0 m/s	20 m/s	10 s	2 m/s²
Sprinter leaving the blocks	0 m/s	10 m/s	2 s	5 m/s²
Stone falling (no air resistance)	0 m/s	29.4 m/s	3 s	9.8 m/s²
Bike braking	8 m/s	0 m/s	4 s	−2 m/s²
Car, 0 to 100 km/h in 5 s	0 m/s	27.8 m/s	5 s	about 5.6 m/s²

Worked example: a car brakes from 30 m/s to a stop in 6 s. a = (0 − 30) ÷ 6 = −5 m/s². The minus sign shows it is slowing down: a deceleration of 5 m/s².

Common mistakes

Doing u − v instead of v − u. Always final minus starting. If the answer is negative, the object is slowing down.
Mixing up velocity and acceleration. Velocity is how fast and in which direction (m/s). Acceleration is how fast the velocity changes (m/s²). A car can be fast with zero acceleration, or stopped for an instant with a big acceleration.
Using km/h. Convert to m/s first by dividing by 3.6: 100 km/h ≈ 27.8 m/s.
Thinking constant speed means no acceleration. Not if the direction changes, as on a bend or a roundabout.
Forgetting the units. m/s² for acceleration, not m/s.

Step by step

Write down u, v and tStarting velocity, final velocity and time. Convert to m/s and seconds.
Find the change in velocityWork out v − u (final minus starting).
Divide by the timeA = (v − u) ÷ t.
Check the signPositive means speeding up; negative means slowing down.
Write the unitM/s².

No time given? Use a = (v² − u²) ÷ 2s with the distance s. Force and mass given? Use a = F ÷ m. To find a final velocity, use v = u + at; to find the time, use t = (v − u) ÷ a.

Practice questions

Acceleration calculations

Try each one, then check your answer.

A toy car speeds up from 0 to 12 m/s in 4 s. Find its acceleration.
Show answer
a = (12 − 0) ÷ 4 = 3 m/s²
A cyclist speeds up from 5 m/s to 25 m/s in 5 s. Find the acceleration.
Show answer
a = (25 − 5) ÷ 5 = 20 ÷ 5 = 4 m/s²
A car slows from 20 m/s to 0 in 5 s. Find its acceleration.
Show answer
a = (0 − 20) ÷ 5 = −4 m/s² (a deceleration of 4 m/s²)
A train slows from 15 m/s to 6 m/s in 3 s. Find its acceleration.
Show answer
a = (6 − 15) ÷ 3 = −9 ÷ 3 = −3 m/s²
A car starts from rest and accelerates at 2 m/s² for 6 s. How fast is it going?
Show answer
v = u + at = 0 + 2 × 6 = 12 m/s
A runner at 10 m/s accelerates at 1.5 m/s² for 4 s. Find her final velocity.
Show answer
v = 10 + 1.5 × 4 = 10 + 6 = 16 m/s
How long does it take to go from 0 to 30 m/s with an acceleration of 3 m/s²?
Show answer
t = (v − u) ÷ a = 30 ÷ 3 = 10 s
A car goes from 0 to 100 km/h in 8 s. Find its acceleration in m/s².
Show answer
100 ÷ 3.6 ≈ 27.8 m/s; a = 27.8 ÷ 8 ≈ 3.5 m/s²
A resultant force of 2,000 N acts on a 1,000 kg car. Find its acceleration.
Show answer
a = F ÷ m = 2,000 ÷ 1,000 = 2 m/s²
A 0.5 kg ball is kicked with a force of 10 N. Find its acceleration.
Show answer
a = 10 ÷ 0.5 = 20 m/s²
What force is needed to accelerate a 1,200 kg car at 3 m/s²?
Show answer
F = m × a = 1,200 × 3 = 3,600 N
A car starts from rest and reaches 20 m/s over a distance of 50 m. Find its acceleration.
Show answer
a = (v² − u²) ÷ 2s = (400 − 0) ÷ 100 = 4 m/s²
A car brakes from 30 m/s to a stop in 45 m. Find its acceleration.
Show answer
a = (0 − 900) ÷ 90 = −10 m/s²
Starting from rest, an object covers 100 m in 10 s with constant acceleration. Find the acceleration.
Show answer
s = ½at², so a = 2s ÷ t² = 200 ÷ 100 = 2 m/s²
A stone is dropped from rest (g = 9.8 m/s², no air resistance). How fast is it falling after 2.5 s?
Show answer
v = 0 + 9.8 × 2.5 = 24.5 m/s
A velocity–time graph is a straight line from 4 m/s at 0 s to 20 m/s at 8 s. What is the acceleration?
Show answer
Gradient = (20 − 4) ÷ 8 = 16 ÷ 8 = 2 m/s²

Word problems

Acceleration in real life

Try each one, then check your answer.

A bus pulls away from a stop and reaches 12 m/s in 8 s. What is its acceleration?
Show answer
a = 12 ÷ 8 = 1.5 m/s²
A cyclist slows from 9 m/s to 3 m/s in 4 s. What is her acceleration?
Show answer
a = (3 − 9) ÷ 4 = −1.5 m/s² (a deceleration of 1.5 m/s²)
A plane starts from rest and reaches 80 m/s in 40 s on the runway. Find its acceleration and the length of runway it uses.
Show answer
a = 80 ÷ 40 = 2 m/s²; s = ½ × 2 × 40² = 1,600 m
An astronaut on the Moon drops a hammer (g = 1.6 m/s²). How fast is it falling after 3 s?
Show answer
v = 1.6 × 3 = 4.8 m/s (on Earth it would be 29.4 m/s)
A 60 kg skater is pushed with a resultant force of 90 N. What is his acceleration?
Show answer
a = 90 ÷ 60 = 1.5 m/s²
A car at 20 m/s brakes with a deceleration of 5 m/s². How long does it take to stop, and how far does it travel while braking?
Show answer
t = 20 ÷ 5 = 4 s; s = (0 − 400) ÷ (2 × −5) = 40 m

Questions people ask

What is the formula for acceleration?

a = (v − u) ÷ t: final velocity minus starting velocity, divided by the time taken. It is also written a = Δv ÷ Δt. With force and mass you can use a = F ÷ m instead.

What is acceleration measured in?

In metres per second squared, m/s². It means "metres per second, every second": an acceleration of 2 m/s² adds 2 m/s to the velocity each second.

What is the difference between velocity and acceleration?

Velocity is how fast something moves and in which direction, in m/s. Acceleration is how quickly that velocity changes, in m/s². A plane cruising at a steady 250 m/s in a straight line has a high velocity but zero acceleration.

What is the acceleration due to gravity?

About 9.8 m/s² (9.81 m/s² more precisely) near the Earth's surface, often rounded to 10 m/s² in school questions. It is the same for every object if there is no air resistance, which is why a hammer and a feather fall together on the Moon.

Can acceleration be negative?

Yes. Negative acceleration means the velocity is decreasing in the positive direction: the object is slowing down (deceleration), or speeding up in the opposite direction.

How do you find acceleration without time?

Use v² = u² + 2as, rearranged to a = (v² − u²) ÷ 2s, where s is the distance. A car going from rest to 20 m/s in 50 m has a = 400 ÷ 100 = 4 m/s².

How do you find acceleration from force and mass?

Use Newton's second law: a = F ÷ m, where F is the resultant force in newtons and m the mass in kilograms. 600 N on 200 kg gives 3 m/s².

What is centripetal acceleration?

It is the acceleration of anything moving in a circle, pointing towards the centre: a = v² ÷ r. A car taking a bend of radius 50 m at 10 m/s has a centripetal acceleration of 100 ÷ 50 = 2 m/s².

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