How To Calculate Acceleration

Acceleration is the rate at which velocity changes with time. It answers a different question from speed. Speed tells you how fast something is going right now; acceleration tells you how quickly that speed is changing, whether it is climbing, falling or reversing. The two are measured in different units, which is the quickest way to keep them apart: speed is metres per second, acceleration is metres per second squared.

The formula is one subtraction and one division. Take the final velocity, subtract the starting velocity, and divide by the time the change took. Written out, that is a equals v minus u over t. The subtraction produces the change in velocity and the division spreads that change evenly over the interval, giving a rate per second.

The subtraction is what makes acceleration a vector and not just a number. If the final velocity is larger than the starting velocity the answer is positive and the object is speeding up in the direction of travel. If the final velocity is smaller the answer is negative, which is what braking looks like. A negative acceleration is not automatically slowing down; it means the acceleration points opposite to whichever direction you called positive.

Units follow from the formula without any extra work. Velocity is measured in metres per second and time in seconds, so dividing one by the other gives metres per second per second, which is written metres per second squared. An acceleration of 4 metres per second squared means the velocity grows by 4 metres per second every second, so after one second the object is moving 4 metres per second, after two seconds 8, and so on.

Acceleration is not the same as speed, and a car at a steady 100 kilometres per hour has an acceleration of zero. That sounds wrong to most people because the engine is clearly working hard, but the engine is working against friction and air resistance rather than changing the speed. Zero acceleration means the velocity is constant, not that nothing is happening.

The average speed over a period of steady acceleration is simply the midpoint of the starting and final velocities. Add them and halve. From rest to 20 metres per second the average is 10 metres per second, and multiplying that by the 5 seconds of the change gives the distance covered. This shortcut works only when the acceleration is constant, which is the case the formula describes.

Multiplying acceleration by mass gives force, which is the second law of motion in its familiar form. A 1200 kilogram car accelerating at 4 metres per second squared needs 4800 newtons of net force. The word net matters, because the engine has to supply that figure on top of whatever is needed to overcome friction and drag, so the force at the wheels is always larger than the calculation suggests.

Free fall is the cleanest case of constant acceleration there is. Near the Earth's surface everything falls with an acceleration of about 9.80665 metres per second squared, regardless of mass, which is why a hammer and a feather drop together in a vacuum. Air resistance is what makes them behave differently in a room, and it is a force, not a change to the acceleration of gravity itself.

Acceleration changes direction as well as magnitude, and a car going round a bend at a constant speed is still accelerating. Its velocity is a vector, so changing the direction of travel changes the velocity even when the number on the speedometer never moves. This centripetal acceleration points towards the centre of the turn and is what keeps the car on the road.

Reading a graph is often faster than using the formula. On a velocity-time graph the acceleration is the slope of the line: a steep line means a large acceleration, a flat line means none, and a line that slopes downwards means negative acceleration. The area under the line gives the distance travelled, which makes the graph a complete description of the motion.