What are speed, velocity and acceleration?

Speed is how far you go each second: speed = distance ÷ time. Its SI unit is the metre per second (m/s). A car that covers 100 m in 10 s is doing 10 m/s. There are 3,600 seconds in an hour and 1,000 metres in a kilometre, so 1 m/s = 3.6 km/h: multiply by 3.6 to get km/h, divide by 3.6 to get back.

Velocity is speed with a direction: 10 m/s forwards is not the same velocity as 10 m/s backwards, even though the speed is the same. On a straight road we use a sign: + for forwards, − for backwards.

Acceleration is how quickly the velocity changes: a = Δv ÷ Δt, in metres per second per second (m/s²). An acceleration of 2 m/s² adds 2 m/s to your velocity every second. Braking is acceleration too, just pointing backwards.

With a steady acceleration, two short rules tell you everything: v = u + a t (your velocity now, from the starting velocity u) and s = u t + ½ a t² (how far from the start you are). On the velocity–time graph, the slope is the acceleration and the area under the line is the distance. Measure distance with the posts, time with the stopwatch, and you have measured all three. Newton's laws (NewtonClear) then tell you what force it takes.

Car makers boast about one number above all: the time from a standstill to 100 km/h. It is really a measure of acceleration. 100 km/h is 27.8 m/s, so a car that takes 10 s gains on average a = Δv ÷ Δt = 27.8 ÷ 10 ≈ 2.8 m/s².

A family hatchback like the one in CarClear takes about 10 s. A high-performance electric car can do it in about 3 s: over 9 m/s², close to 1 g, the acceleration of a falling stone. Electric motors give full pull from standstill, and with all four wheels driven the tyres' grip is the only limit.

Watch the curves on the board: nobody accelerates steadily. At first grip sets the limit, then power does: the push a motor can give is P ÷ v, so it fades as speed grows, while air drag grows with v². That is why the lines bend over. A 150 cc motorcycle (MotorcycleClear) starts as briskly as the hatchback but runs out of power, and a strong cyclist (CycleClear) tops out around 40 km/h.

Acceleration is often quoted in g: divide by 9.81 m/s². The hatchback's 2.8 m/s² is about 0.3 g, a gentle push into the seat.

Braking is acceleration backwards. A car on a dry road can slow at about 6.5 m/s², losing 6.5 m/s of speed every second. But before the brakes even start, the driver has to notice and react. That takes about 1.5 s for most people, longer if they are looking at a phone.

So a stop comes in two parts. The thinking distance is speed × reaction time: at 50 km/h (13.9 m/s) that is about 21 m, before the car slows at all. The braking distance is v² ÷ 2a, and that square matters: double the speed and it takes four times as far. Together: about 18 m from 30 km/h, 36 m from 50 km/h and 71 m from 80 km/h on a dry road. On the speed–time graph the two parts are the two shaded areas.

That is why many cities set 30 km/h near schools. A child who runs out 25 m ahead is safe from a car at 30 km/h. At 50 km/h the car is still doing about 40 km/h when it gets there. The WHO notes that a pedestrian's risk of dying rises about 4.5 times between being hit at 50 km/h and at 65 km/h, and it recommends 30 km/h where people walk, cycle and play.

Falling is the purest acceleration there is. Near the Earth, anything dropped gains 9.81 m/s every second: g = 9.81 m/s². After 1 s it falls at 35 km/h, after 2 s at 71 km/h. The strobe copies of the ball get further apart each 0.2 s, just like the dots in chapter 1.

In a lift, scales measure how hard the floor pushes you: m (g + a). Speeding up on the way up, you feel heavier; on a drop-tower ride in free fall the scales read zero, and you feel weightless. "g-force" is this push compared with your normal weight. See NewtonClear for the forces behind it.

A fan runs at a steady speed. Is anything accelerating? Yes: the tip of every blade. Its speed never changes, but its direction does, all the time, so its velocity changes. A change in velocity is acceleration, and for something going round a circle it points to the centre: a = v² ÷ r.

The tip of a 400 mm desk fan (FanClear) at 1,300 rpm goes round a 0.2 m circle 22 times a second: v = 2π r n ÷ 60 ≈ 27 m/s, nearly 100 km/h. Its acceleration towards the hub is about 3,700 m/s², around 380 g. The blade has to pull that hard on its tip to keep it going round.

In a washing machine (WasherClear) the drum tumbles slowly, about 50 rpm, so the clothes are lifted and then fall: the acceleration needed, 0.7 g, is less than gravity. At 1,200 rpm the drum wall pushes on the clothes with about 400 g, but the water can't follow the curve: it flies straight on through the holes. The same maths turns a car or bike round a bend (see NewtonClear).

A car's speedometer shows the instantaneous speed: how fast you are going right now. The trip computer shows the average speed: total distance ÷ total time, red lights and all. Take the average over a shorter and shorter time and it gets closer and closer to the needle. That idea, the rate over a vanishingly small time, is where calculus began.

Speed is not the same as velocity. Run one lap of a 400 m track at 5 m/s and you have covered 400 m in 80 s: an average speed of 5 m/s. But you finish where you started, so your displacement is zero and your average velocity is zero too. On the bends your speed stays the same while your velocity turns, so you are accelerating even at a steady pace.

Myth: heavier things fall faster. Aristotle taught it, and a feather seems to prove it. But drop a bowling ball and a tennis ball together from 20 m and they land within a blink of each other. Gravity gives every object the same acceleration, 9.81 m/s². Only air resistance makes the difference, and it matters most for light, fluffy things. On the Moon in 1971, astronaut David Scott dropped a hammer and a feather, and they landed together.

Velocity depends on who is watching. Walk forwards at 5 km/h on a train doing 100 km/h: to your friend on the train you do 5 km/h, to someone on the platform 105 km/h. Every velocity is measured relative to something, its frame of reference. Even the platform is spinning with the Earth at about 1,670 km/h at the equator.

For everyday speeds you just add. But near the speed of light, c = 299,792,458 m/s, adding stops working. Einstein showed in 1905 that velocities combine as (u + v) ÷ (1 + u v ÷ c²). Fire a probe at 0.8 c from a ship doing 0.8 c and it goes 0.98 c, not 1.6 c. Nothing with mass can reach c.