Acceleration

An object accelerates when its velocity changes. The change may be in its speed, its direction of motion, or both. Acceleration describes how quickly velocity changes.

Average acceleration

Over a time interval, average acceleration is the change in velocity divided by the time taken:

Here, $$a$$ is average acceleration, $$\Delta v$$ is the final velocity minus the initial velocity, and $$\Delta t$$ is the elapsed time. Acceleration is measured in metres per second squared, written $$\mathrm{m/s^2}$$.

Acceleration is a vector, so it has a direction as well as a magnitude. For straight-line motion, choose one direction as positive; acceleration in the opposite direction is negative. A negative acceleration does not always mean that an object is slowing down.

For example, suppose right is positive and an object’s velocity changes from $$+6\ \mathrm{m/s}$$ to $$+2\ \mathrm{m/s}$$ in $$2\ \mathrm{s}$$. Its average acceleration is $$(-4\ \mathrm{m/s})/(2\ \mathrm{s}) = -2\ \mathrm{m/s^2}$$. The acceleration is to the left, opposite to the object’s velocity, so the object slows down.

Acceleration and direction of motion

In straight-line motion, the effect on speed depends on how the direction of acceleration compares with the direction of velocity:

• Acceleration in the same direction as velocity increases speed.
• Acceleration in the opposite direction to velocity decreases speed.

If acceleration continues opposite to the motion until the object stops, it can then make the object move in the opposite direction. Acceleration can therefore change the direction of motion as well as the speed.

Uniform and non-uniform acceleration

Uniform acceleration is constant: velocity changes by equal amounts in equal time intervals. Non-uniform acceleration changes with time, so the change in velocity need not be equal in equal time intervals.

Falling motion and terminal velocity

When an object falls, its weight pulls it downwards. Whether its downward velocity keeps increasing depends on the other forces acting on it.

Falling without resistance

In a uniform gravitational field, an object falling without air or liquid resistance accelerates downwards at the constant gravitational field strength, $$g$$. Its downward velocity increases by equal amounts in equal time intervals.

Falling through air or a liquid

Air or liquid resistance acts upwards on a falling object, opposing its motion. As the object speeds up, resistance generally increases. This reduces the resultant downward force, so the object continues to speed up but its acceleration decreases.

The air or liquid can also exert an upward force called upthrust. When the upward forces, including resistance and any upthrust, balance the object’s weight, the resultant force is zero. The object has then reached terminal velocity.

At terminal velocity, acceleration is zero. The object does not stop: it continues falling at a constant velocity.