The acceleration formula

The acceleration formula is a = (v − u) / t: acceleration equals the change in velocity divided by the time it takes. Here u is the starting velocity, v is the final velocity and t is the time. In the US it is usually written a = Δv/Δt or a = (v_f − v_i)/t. Acceleration is measured in metres per second squared, m/s², so an acceleration of 3 m/s² means the velocity goes up by 3 m/s every second.

The formula for acceleration

Strictly, this gives the average acceleration over the time t. If the acceleration is steady, the average and the actual acceleration are the same. If it varies, the formula tells you the average over that stretch of time.

Rearranging the acceleration equation

Acceleration units

The unit of acceleration is m/s², read as metres per second per second. That is exactly what acceleration is: a change in velocity (m/s) for every second that passes. In US customary units you may see ft/s². Near Earth's surface, things fall with an acceleration of about 9.8 m/s², or 32 ft/s².

How to calculate acceleration

1. Write down u, v and t from the question, in m/s and s. "Starts from rest" means u = 0; "comes to a stop" means v = 0.
2. Subtract to find the change in velocity, v − u. Final minus initial, always in that order.
3. Divide by the time.
4. Give the unit, m/s². A negative answer means the object is slowing down, if it is moving in the positive direction.

Example 1: a car from 0 to 60 mph

A car accelerates from rest to 60 mph in 8.0 s. What is its average acceleration?

Convert to SI units first: 60 mph is 26.8 m/s. Then:

The car's average acceleration is 3.35 m/s², about a third of the acceleration due to gravity.

Example 2: deceleration

A cyclist slows from 12 m/s to 4 m/s in 4 s. Find the acceleration.

The acceleration is −2 m/s². The minus sign shows the cyclist is slowing down. You can also say the deceleration is 2 m/s²: deceleration is the size of a negative acceleration, so it is given without the minus sign.

Example 3: finding the final velocity

A ball rolling at 1.5 m/s speeds up at 0.4 m/s² for 5 s. How fast is it going at the end?

Example 4: finding the time

A train starts from rest and accelerates at 0.5 m/s². How long does it take to reach 20 m/s?

Acceleration from a velocity-time graph

On a velocity-time graph, acceleration is the gradient (slope) of the line. A steeper line means a bigger acceleration, a flat line means zero acceleration, and a line sloping down means the object is slowing down.

Here x is time in seconds and y is velocity in m/s. For the first 4 s the velocity rises from 6 m/s to 18 m/s, so the gradient is (18 − 6) ÷ 4 = 3 m/s². After that the line is flat: the velocity is constant and the acceleration is zero. The area under the line is the distance travelled. Try reading one yourself, then press Mark to check your answer in the course.

Other acceleration formulas

The basic formula needs a time. When a question gives you something else, one of these will fit.

Acceleration formula with distance

If you know the two velocities and the distance but not the time, rearrange v² = u² + 2as, one of the SUVAT equations:

Example 5: a plane starts from rest and reaches its take-off speed of 75 m/s after 1500 m of runway. Its acceleration is 75² ÷ (2 × 1500) = 5625 ÷ 3000 = 1.875 m/s².

Acceleration from force and mass

Newton's second law links acceleration to the resultant force F on a mass m:

Example 6: a resultant force of 3000 N acts on a 1200 kg car. Its acceleration is 3000 ÷ 1200 = 2.5 m/s². Use the resultant force: the driving force minus friction and air resistance, not the driving force alone. The Force Calculator works this out for you.

Acceleration due to gravity formula

Near Earth's surface, g is about 9.8 m/s² (9.81 m/s² to 3 significant figures). For any planet, the acceleration due to gravity at its surface comes from Newton's law of gravitation, where G is the gravitational constant, M the planet's mass and r its radius:

With Earth's mass, 5.97 × 10²⁴ kg, and mean radius, 6371 km, this gives 9.81 m/s². NASA lists Earth's surface gravity as 9.8 m/s². The real value varies slightly from place to place, because Earth spins and is not a perfect sphere.

Centripetal acceleration formula

Anything moving in a circle is accelerating, even at a steady speed, because its direction keeps changing. The acceleration points towards the centre of the circle:

where v is the speed, r is the radius and ω is the angular velocity in rad/s. Example 7: a car takes a bend of radius 50 m at 15 m/s. Its centripetal acceleration is 15² ÷ 50 = 4.5 m/s², towards the centre of the bend. The Circular Motion Calculator shows this for any speed and radius.

Angular and tangential acceleration formulas

For something spinning, angular acceleration α is the change in angular velocity divided by time, and tangential acceleration is the linear acceleration of a point at radius r:

Example 8: a wheel of radius 0.3 m speeds up from rest to 12 rad/s in 3 s. Its angular acceleration is 12 ÷ 3 = 4 rad/s², and a point on its rim has a tangential acceleration of 0.3 × 4 = 1.2 m/s².

Magnitude of acceleration

Acceleration is a vector, so it has a direction as well as a size. If an object accelerates at 3 m/s² east and 4 m/s² north at the same time, the magnitude of its acceleration is found with Pythagoras: √(3² + 4²) = 5 m/s².

Which equation can be used to solve for acceleration?

Pick the one that matches what the question gives you:

Common mistakes

• Subtracting the wrong way. Change in velocity is final minus initial, v − u. Getting it backwards flips the sign.
• Using speed in km/h or mph. Convert to m/s first: divide km/h by 3.6, or multiply mph by 0.447.
• Confusing acceleration with velocity. A fast car on cruise control has zero acceleration. A car just starting from rest can have a large one.
• Thinking constant speed means zero acceleration. In a circle, the direction changes, so there is a centripetal acceleration.
• Using the driving force instead of the resultant force in a = F/m.
• Leaving out the unit or writing m/s instead of m/s².

Is the acceleration formula on the formula sheet?

Yes, on the sheets checked in November 2026. AQA's GCSE Physics equations sheet gives acceleration = change in velocity ÷ time taken, a = Δv/t, and v² − u² = 2as. AP Physics 1 gives v = v₀ + at on its reference sheet, which rearranges to the same formula. You still need to know when to use which, and the units.

Practise motion in a short timed test

Acceleration is in nearly every motion question. High School Physics and AP Physics 1 each give you a short timed practice test every day, with 8 questions worth 25 to 35 points. In the UK, the same practice opens as GCSE Physics Higher. To build up from the start, the Laws of Motion course begins with velocity and acceleration and works up to Newton's laws and projectiles.

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