How to calculate acceleration:
from a change in speed over time

Acceleration is how fast velocity changes

If a vehicle is doing 5 m/s and, 4 seconds later, 25 m/s, its acceleration is 5 m/s². That figure means the speed climbs by 5 metres per second during every second that passes. The way to calculate acceleration is always the same: take the change in velocity and divide it by the time over which the change happened.

The unit tells the story. Metres per second is a speed; dividing by seconds once more gives metres per second per second, written m/s². A value of 5 m/s² is not a large number on its own, but it describes a sports car pulling away far better than a top speed does, because it captures how quickly the speed builds.

Acceleration is a vector, so it has a direction as well as a size. Speeding up in the direction you are travelling gives a positive value. Slowing down gives a negative value, which many people call deceleration. Turning at a constant speed is also acceleration, because the direction of the velocity keeps changing, though that case needs a different formula.

The formula and a first worked sum

Rearranging the same relationship answers other questions. Time needed to reach a target speed is t = (v − u) ÷ a, and the speed you will have after a time is v = u + a × t. At 5 m/s² from rest, you reach 30 m/s in 6 seconds.

Converting km/h before you divide

Car makers quote performance in kilometres per hour, but acceleration in m/s² needs speed in m/s. Divide km/h by 3.6 to convert. Forgetting this step is the most common reason a hand calculation comes out wrong by a factor of more than three.

Take a hatchback that goes from 0 to 100 km/h in 8 seconds. Convert first: 100 ÷ 3.6 = 27.78 m/s. Then divide by the time: 27.78 ÷ 8 = 3.47 m/s². Comparing that with the pull of gravity, 9.81 m/s², gives about 0.35 g, which is roughly what an occupant feels pressing them into the seat.

Slowing down: negative acceleration and stopping distance

The formula does not change when a vehicle brakes; the final velocity is simply smaller than the initial one, so the answer turns negative. A car moving at 20 m/s that comes to rest in 5 seconds has a = (0 − 20) ÷ 5 = −4 m/s².

Once you know the braking rate, the distance covered while stopping follows from s = u² ÷ (2 × a), using the size of a. With u = 20 m/s and a = 4 m/s², s = 400 ÷ 8 = 50 m. Because speed is squared, doubling the speed to 40 m/s at the same braking rate needs 200 m, four times as far, not twice.

Other ways to reach the same number

When time is not given, other relationships fill the gap. Newton's second law gives a = F ÷ m, so a net force of 3,000 N on a 1,000 kg car produces 3 m/s². When distance is known instead of time, use a = (v² − u²) ÷ (2 × s). A cyclist who goes from 4 m/s to 10 m/s over 21 m is accelerating at (100 − 16) ÷ 42 = 2 m/s².

All of these assume the acceleration stays constant during the interval. Real vehicles do not do that: engine torque falls as speed rises, so the first second of a launch is usually stronger than the last. Treat the result as an average unless you have data taken at short intervals.

Benchmarks for judging whether an answer is plausible

A result is easier to trust when you know roughly where it should land. Most everyday movements sit well below the pull of gravity, and anything above about 4 g is the territory of fighter jets, rocket launches and crash tests.

The figures in the middle of the table follow directly from the km/h conversion above: 27.78 m/s divided by 12 s is 2.31 m/s², and divided by 10 s is 2.78 m/s². If your own sum for an ordinary road car lands near 30 m/s², the likeliest cause is a missed conversion or a time entered in the wrong unit.

Collisions are the extreme case. A ball arriving at a bat at 12 m/s and leaving in the opposite direction at 18 m/s, with a contact time of 0.02 s, has a change in velocity of −18 − 12 = −30 m/s, so a = −30 ÷ 0.02 = −1,500 m/s². The sign shows the direction flipped, and the size shows how violent a short contact time makes the acceleration.

Slips that change the answer

• Mixing units, for example dividing km/h by seconds and calling the result m/s².
• Using total speed instead of the change in speed when the object was already moving.
• Dropping the sign. A result of −4 m/s² is braking, not an error.
• Reading average acceleration over 10 seconds as if it were the peak value at launch.
• Treating g as a unit of speed. One g is an acceleration of 9.81 m/s².

When you need to double-check a hand calculation or run several speed and time combinations quickly, a calculator is faster, but keep the units explicit so you can spot an impossible answer before using it.