Velocity and Acceleration

You already know about speed — how many metres something covers each second. But speed on its own leaves out one big question: which way? A car doing 30 m/s straight towards you and a car doing 30 m/s straight away from you are very different things, even though the speedometer reads the same in both.

So physicists add the missing piece. Velocity is speed together with a stated direction. "30 m/s" is a speed. "30 m/s due north" is a velocity. The number tells you how fast; the direction tells you where to.

Same speed, different velocity

Because velocity carries a direction, two things can share one speed and still have completely different velocities. Picture two cars on a motorway, each at exactly 40 mph — one heading north, one heading south. Same speed, opposite directions, so their velocities are different.

A quantity that needs a direction like this is called a vector. Velocity is a vector; plain speed (just a number, no arrow) is a scalar. Change the size or the direction and you have changed the velocity — even if the number of metres per second stays exactly the same. Hold on to that last idea; it is the secret behind the next section.

These three trip up almost everyone:

Speed and velocity are not the same word twice. Speed is just the number; velocity is the number plus a direction. A car can travel at a steady speed while its velocity keeps changing.
Going round a bend at a steady speed, you are still accelerating. Your speed isn't changing, but your direction is — so your velocity is changing, and any change of velocity is an acceleration. That's why you feel thrown to one side on a roundabout.
"Decelerating" is not a different thing from accelerating. Slowing down is simply a negative acceleration. There is one idea here, not two.

Acceleration: how fast velocity changes

Acceleration tells you how quickly an object's velocity is changing. If a car's velocity climbs from a starting value u to a final value v in a time t, its acceleration is the change in velocity shared out over the time it took:

a = \dfrac{v - u}{t}

Velocity is measured in metres per second (m/s) and time in seconds (s), so acceleration is measured in metres per second, per second — written \text{m/s}^2. An acceleration of 2\ \text{m/s}^2 means the velocity grows by 2 m/s during every single second: after one second it is 2 m/s faster, after two seconds 4 m/s faster, and so on.

Worked examples

1. A car pulling away. It goes from rest (u = 0) to v = 30\ \text{m/s} in t = 6\ \text{s}.

a = \dfrac{v - u}{t} = \dfrac{30 - 0}{6} = 5\ \text{m/s}^2.

2. A sprinter off the blocks. From rest to v = 10\ \text{m/s} in t = 2\ \text{s}.

a = \dfrac{10 - 0}{2} = 5\ \text{m/s}^2.

3. Braking (a negative acceleration). A car slows from u = 20\ \text{m/s} to a stop (v = 0) in t = 4\ \text{s}. Because the velocity falls, the answer comes out negative:

a = \dfrac{0 - 20}{4} = -5\ \text{m/s}^2.

The minus sign isn't a mistake to fix — it is the information. A positive acceleration means the velocity is growing; a negative one (a deceleration) means it is shrinking.

Three ways to accelerate

Since acceleration is any change of velocity, there are three different situations that all count as accelerating:

Speeding up in a straight line — the velocity grows, so the acceleration is positive.
Slowing down — the velocity shrinks, so the acceleration is negative.
Changing direction at a steady speed — the size of the velocity stays put but its direction turns, so the velocity still changes. Rounding a corner, orbiting, swinging on a rope: all accelerations.

Only one kind of motion has zero acceleration: travelling in a perfectly straight line at a perfectly steady speed. The moment either the speed or the direction changes, you are accelerating.

Everyday accelerations

It helps to have a feel for the numbers. Here are some rough values, all in \text{m/s}^2:

A family car pulling away gently: about 2\text{–}3\ \text{m/s}^2.
A sprinter exploding out of the blocks: about 5\ \text{m/s}^2.
Anything dropped near the Earth (gravity): about 10\ \text{m/s}^2.
A quick sports car (0 to 60 mph in ~4 s): about 7\ \text{m/s}^2.
Slamming the brakes on: around -8\ \text{m/s}^2 (a big negative).

When you accelerate hard, your own body notices — that shove into the seat as a car launches, or the lift in your stomach on a rollercoaster drop. Engineers measure that squeeze in "g", where 1g is the everyday 10\ \text{m/s}^2 of gravity. A fast road car corners at about 1g; a Formula 1 car brakes and turns at up to 5g; a fighter pilot pulling a tight turn can hit 9g — nine times gravity — and has to squeeze their legs and grunt to keep the blood from draining out of their head. A rocket crew heading to orbit rides at around 3g for minutes on end, pinned to their couches too heavy to lift an arm. Same physics as a car pulling away — just turned up loud.

Try it: build an acceleration

Set a start velocity u, an end velocity v and the time t below. The two arrows show the starting and ending velocities of the car, and the readout works out the acceleration a = \tfrac{v-u}{t} live. Make v bigger than u for a positive acceleration, smaller for a negative one, and equal for no acceleration at all.