The Acceleration Formula, Explained with Examples
By ToolNimba Editorial Team June 20, 2026 7 min read
Quick answer
Average acceleration = (final velocity minus initial velocity) divided by time, written as a = (v minus u) / t. Here v is the final velocity, u is the starting velocity, and t is the time taken. Acceleration is measured in metres per second squared (m/s squared).
Acceleration tells you how quickly an object's velocity is changing. A car that goes from a standstill to highway speed in a few seconds has a large acceleration; a car easing up to the same speed over a full minute has a small one. In this guide you will see the core formula, the units that go with it, how acceleration relates to velocity, and several worked examples you can follow step by step.
What is acceleration?
Acceleration is the rate at which an object's velocity changes over time. Because velocity has both a size and a direction, acceleration is a vector quantity too. Any time an object speeds up, slows down, or changes direction, it is accelerating. Slowing down is sometimes called deceleration, but in physics it is simply acceleration with the opposite sign to the motion.
The key idea is change in velocity. If an object's velocity is not changing, its acceleration is zero, even if it is moving very fast. A train cruising at a steady 200 km/h in a straight line has zero acceleration. The moment it brakes or speeds up, acceleration appears. To dig into velocity itself first, see the velocity formula guide.
The average acceleration formula
The most common form of the acceleration formula is the average acceleration over an interval of time:
Average acceleration
a = (v minus u) / t, or in words, acceleration = (final velocity minus initial velocity) divided by the time elapsed.
Each symbol has a clear meaning. The letter a is the acceleration you are solving for, v is the final velocity at the end of the interval, u is the initial velocity at the start, and t is the time it took to go from u to v. The top of the fraction, v minus u, is the change in velocity, often written as the Greek letter delta v.
Symbols in the acceleration formula
| Symbol | Meaning | Typical unit |
|---|---|---|
| a | Acceleration | m/s squared |
| v | Final velocity | m/s |
| u | Initial velocity | m/s |
| t | Time elapsed | seconds |
| v minus u | Change in velocity | m/s |
Units of acceleration
Because acceleration is a velocity (in metres per second) divided by a time (in seconds), its SI unit is metres per second per second, written as metres per second squared (m/s squared). You can read it as "how many metres per second the velocity gains every second". An acceleration of 3 m/s squared means the velocity increases by 3 m/s during each second that passes.
Other units appear depending on the field. Vehicle testers often quote acceleration in km/h per second, and physicists comparing forces sometimes use g, the acceleration due to gravity near Earth's surface, which is about 9.8 m/s squared. As long as the top of the fraction is a velocity and the bottom is a time, you have a valid acceleration unit. If you need to switch distance units first, a length converter handles that step.
Common acceleration reference values
| Situation | Approximate acceleration |
|---|---|
| Gravity near Earth's surface | 9.8 m/s squared |
| Family car, gentle start | 2 to 3 m/s squared |
| Sports car, hard launch | 8 to 10 m/s squared |
| Commercial jet at takeoff | 3 to 4 m/s squared |
| Emergency car braking | negative 8 m/s squared |
Types of acceleration
Acceleration shows up in several forms depending on whether the change is steady, and on whether the motion is along a straight line or around a curve. Knowing which type you are dealing with tells you which formula to reach for.
Common types of acceleration
| Type | What it means | When you use it |
|---|---|---|
| Average acceleration | Total change in velocity over the whole time interval | Most everyday problems and the a = (v minus u) / t formula |
| Instantaneous acceleration | Acceleration at a single moment in time | When acceleration is changing from instant to instant |
| Uniform acceleration | Velocity changes by equal amounts in equal time intervals | Free fall and the standard equations of motion |
| Non-uniform acceleration | Velocity changes by unequal amounts in equal intervals | Real engines, drag, and varying forces |
| Centripetal acceleration | Acceleration pointing toward the centre of a circular path | Anything moving in a circle at constant speed |
| Deceleration | Acceleration directed opposite to the motion | Braking and slowing down |
For straight-line motion at a steady rate, average and instantaneous acceleration are identical, so the simple a = (v minus u) / t formula is all you need. The moment acceleration starts varying, or the path curves, you switch to instantaneous values or to the circular-motion formulas instead.
Worked example: an accelerating car
A car speeds up from 10 m/s to 30 m/s in 5 seconds. Find its average acceleration.
- List the values: u = 10 m/s, v = 30 m/s, t = 5 s.
- Find the change in velocity: v minus u = 30 minus 10 = 20 m/s.
- Apply the formula: a = (v minus u) / t = 20 / 5.
- Divide: 20 divided by 5 equals 4.
- State the answer with units: the acceleration is 4 m/s squared.
The positive sign tells you the car is speeding up in the direction it is already moving. Each second, its velocity climbs by another 4 m/s.
Worked example: braking (negative acceleration)
A cyclist slows from 12 m/s to 0 m/s in 4 seconds while braking. Find the acceleration.
- List the values: u = 12 m/s, v = 0 m/s, t = 4 s.
- Find the change in velocity: v minus u = 0 minus 12 = negative 12 m/s.
- Apply the formula: a = (v minus u) / t = negative 12 / 4.
- Divide: negative 12 divided by 4 equals negative 3.
- State the answer: the acceleration is negative 3 m/s squared.
The negative sign means the acceleration points opposite to the motion, which is exactly what slowing down looks like. The size of the change, 3 m/s squared, is sometimes called the deceleration.
Rearranging the formula
The same equation answers other questions once you rearrange it. If a problem gives you the acceleration and asks for a missing velocity or time, solve for the unknown instead of for a.
- To find the final velocity: v = u + a x t. Multiply the acceleration by the time and add the starting velocity.
- To find the initial velocity: u = v minus a x t. Subtract the gained velocity from the final velocity.
- To find the time taken: t = (v minus u) / a. Divide the change in velocity by the acceleration.
These are all the same relationship seen from different angles. The form v = u + a x t is one of the standard equations of motion and is covered further in the velocity formula guide.
The equations of motion (kinematic equations)
When acceleration is constant, three equations of motion connect velocity, time, acceleration, and displacement. They all build on the same definition of acceleration and let you solve for whatever quantity a problem leaves out. Here s is the displacement, the distance travelled in a given direction.
The three equations of motion for constant acceleration
| Equation | What it links | Use it when |
|---|---|---|
| v = u + a x t | Velocity, acceleration, time | You know the time and want the final velocity |
| s = u x t + (1/2) x a x t squared | Displacement, time, acceleration | You want the distance covered over a time |
| v squared = u squared + 2 x a x s | Velocity, acceleration, displacement | You have a distance but no time value |
These hold only while acceleration stays constant. If the acceleration changes during the motion, you have to break the journey into intervals or use calculus instead.
Acceleration and force: a = F / m
Acceleration also comes straight from Newton's second law. Rearranging F = m x a gives a = F / m, which says the acceleration of an object equals the net force on it divided by its mass. A larger force gives more acceleration, while a heavier object accelerates less for the same push.
Suppose a net force of 50 newtons acts on a 10 kg cart. The acceleration is a = F / m = 50 / 10 = 5 m/s squared. To work the other way and find the force, multiply mass by acceleration; a multiplication-style calculation keeps the arithmetic tidy when the numbers get awkward.
Acceleration, velocity, and rate of change
Velocity is how fast position changes, and acceleration is how fast velocity changes. That makes acceleration a "rate of change of a rate of change". On a velocity-versus-time graph, the acceleration is simply the slope of the line, the same way the slope formula measures steepness on any straight line. A steeper line means a larger acceleration, and a flat line means zero acceleration.
This connection to slope is why acceleration shares so much with the average rate of change you meet in algebra. Both divide a change in one quantity by the change in another, and both can be positive, negative, or zero depending on the direction of that change.
Common mistakes to avoid
- Forgetting to subtract. The top of the fraction is the change in velocity, v minus u, not just the final velocity.
- Dropping the sign. A negative acceleration is a real, meaningful answer that signals slowing down or motion in the opposite direction.
- Mixing units. Keep velocity in m/s and time in seconds before dividing, or convert first so the units match.
- Confusing high speed with high acceleration. An object can move very fast while having zero acceleration if its velocity is steady.
- Using the wrong unit. Acceleration is measured in metres per second squared, not metres per second; the squared second is what separates it from velocity.
Good to know
Average acceleration covers a whole interval, while instantaneous acceleration is the acceleration at a single moment. The two are equal only when acceleration is constant. Acceleration is also the link between motion and force: Newton's second law, F = m x a, says the force on an object equals its mass times its acceleration, which is why heavier objects need more force to speed up at the same rate.
Frequently asked questions
What is the acceleration formula?
The average acceleration formula is a = (v minus u) / t, meaning acceleration equals the final velocity minus the initial velocity, divided by the time taken. The result is measured in metres per second squared.
What are the units of acceleration?
The SI unit of acceleration is metres per second squared (m/s squared). It represents how many metres per second the velocity changes during each second. Other units, such as km/h per second or g (about 9.8 m/s squared), are also used in specific fields.
How do you calculate acceleration?
Subtract the initial velocity from the final velocity to get the change in velocity, then divide by the time elapsed. For example, a car going from 10 m/s to 30 m/s in 5 seconds has an acceleration of (30 minus 10) / 5 = 4 m/s squared.
Can acceleration be negative?
Yes. A negative acceleration means the velocity is decreasing, or that the object is accelerating in the direction you defined as negative. It often describes braking or slowing down and is sometimes called deceleration.
Is acceleration the same as velocity?
No. Velocity is how fast position changes, while acceleration is how fast velocity changes. An object can have a high velocity but zero acceleration if its velocity is steady. Velocity is measured in m/s and acceleration in m/s squared.
What is the difference between acceleration and speed?
Speed measures how fast an object moves, while acceleration measures how quickly that motion changes. A car at a constant 60 km/h has speed but zero acceleration. Only when its speed or direction changes does it accelerate.
What are the three equations of motion?
For constant acceleration, the three equations of motion are v = u + a x t, s = u x t + half a x t squared, and v squared = u squared + 2 x a x s. They link velocity, acceleration, time, and displacement so you can solve for whichever quantity a problem leaves out.
How do you find acceleration from force and mass?
Use Newton's second law rearranged as a = F / m. Divide the net force in newtons by the mass in kilograms to get acceleration in metres per second squared. For example, a 50 N force on a 10 kg object gives a = 50 / 10 = 5 m/s squared.
What is centripetal acceleration?
Centripetal acceleration is the acceleration of an object moving in a circle, always pointing toward the centre of the path. Even at constant speed, the direction keeps changing, so there is acceleration. It is found with a = v squared / r, where r is the radius.