How things move

Speed says how fast and nothing about where to. Two cars leave the same junction at 50 kilometres an hour, one heading north and one heading south. They have the same speed, yet in an hour they will be 100 kilometres apart. To say where a moving thing will end up you need its speed together with its direction, and that pair is its velocity. Fifty kilometres an hour is a speed. Fifty kilometres an hour northward is a velocity.

The difference matters whenever something turns. A car going round a roundabout at a steady 30 kilometres an hour has a constant speed, as its speedometer shows. Its velocity is not constant, because its direction keeps changing all the way round. In physics, a change of direction counts as a change in velocity even when the speed holds still.

Walk 3 kilometres east along a straight road, then turn round and walk 3 kilometres back. You have covered a distance of 6 kilometres, and your legs know it. But you are standing where you started. Your displacement, the straight-line change in your position from start to finish, is zero.

Distance and displacement are examples of two kinds of quantity. A quantity that has only a size, such as distance, speed or temperature, is called a scalar. One that has both a size and a direction, such as displacement, velocity or force, is called a vector, and it is often drawn as an arrow whose length shows the size. The pairing runs through the whole subject: speed is built from distance, and velocity is built from displacement.

Things rarely move at one fixed velocity. A car pulls away from the lights, a ball slows as it rolls uphill, a bus swings round a corner. Each of these is a change in velocity, and the rate at which velocity changes is called acceleration. A car that goes from rest to 30 metres per second in 10 seconds gains 3 metres per second every second, and that is its acceleration.

In everyday speech, accelerating means speeding up. In physics the word covers any change in velocity. A braking car is accelerating, because its velocity is changing, only in the direction opposite to its motion. A car rounding a bend at a steady speed is accelerating too, since its direction is changing. Only something moving in a straight line at a constant speed has no acceleration at all.

A picture often shows motion better than a list of numbers. Plot time along the bottom and speed up the side and you have a speed–time graph, and its shape tells the story at a glance.

A car cruising at a constant speed draws a flat line: the speed is not changing, so the acceleration is zero. A car gaining speed steadily draws a straight line that rises, and the steeper the line, the greater the acceleration. A line that falls shows a car slowing down, and a curve shows an acceleration that is itself changing.

Reading these shapes was one of the first skills physics needed, because a steady acceleration, the straight rising line, turns out to describe everything that falls.

For nearly two thousand years scholars followed the Greek philosopher Aristotle, who taught that heavy things fall faster than light ones. Few put it to a careful test. A falling stone is too quick to time by eye, and there were no stopwatches.

In the early 1600s the Italian scientist Galileo Galilei found a way round the problem. He rolled bronze balls down a gently sloping ramp. Rolling is falling in slow motion: the ramp slowed the fall down enough for him to time it. He found a simple pattern. In twice the time, a ball rolled four times as far. In three times the time, nine times as far. The distance grew with the square of the time. That is exactly what a steady acceleration produces, and Galileo concluded that a falling body gains speed at a constant rate.

Galileo’s second conclusion was stranger. Take away the air, he argued, and every object falls in exactly the same way, whatever it weighs. A cannonball and a pebble dropped together would land together.

Falling with nothing but gravity acting is called free fall, and near the Earth’s surface it always means the same acceleration: a falling object gains about 10 metres per second of speed every second. The pull never switches off. A ball thrown straight up loses speed at the same rate, and even at the very top, where it hangs motionless for an instant, gravity is still accelerating it downward.

In 1971 the astronaut David Scott tried the experiment on the airless Moon. He let go of a hammer and a feather together, and they landed at the same time.

In a room, a feather does not keep up with a hammer. Gravity accelerates both alike, but the air pushes back on anything moving through it, and that push is called air resistance. On a compact, heavy hammer it hardly matters. On a light, fluffy feather it is nearly as strong as the feather’s weight, so the feather drifts down.

Air resistance grows with speed. A skydiver who steps out of a plane speeds up at first, as free fall predicts. But the faster the fall, the harder the air pushes back, until the upward push equals the skydiver’s weight. The two forces now balance, and the skydiver stops gaining speed. The rest of the fall happens at a steady speed called the terminal velocity, about 200 kilometres an hour for a skydiver falling belly-down.

Slide a book across a table and it soon stops. From this Aristotle concluded that a moving thing needs a constant push. A thrown stone flies on after leaving the hand, which he supposed the air kept pushing along.

Around 1020 the Persian scholar Ibn Sīnā gave a better account. The thrower, he wrote, gives the stone an inclination to keep moving, and only something resisting it, such as the air, uses that inclination up. In a void, he reasoned, the stone would never stop, which he took as a sign that no true void exists. Galileo argued the same for a ball on a smooth, level surface, and Isaac Newton made it a law: a moving object keeps moving in a straight line at a steady speed unless a force acts on it. This persistence is called inertia. The book stops because a force, friction, is acting on it.

Sit on a smooth-flying aeroplane with a cup of coffee on the tray. To you the cup is at rest. To someone on the ground it is crossing the sky at 900 kilometres an hour. Neither of you is wrong. Every speed is measured relative to something, and the viewpoint a motion is measured from is called a frame of reference.

Change the frame and the numbers change. A train passes through a station at 30 metres per second, and a passenger walks forward along the aisle at 1 metre per second. In the train’s frame the passenger’s speed is 1. In the station’s frame the two speeds add, and the passenger passes the platform at 31 metres per second. Walking toward the back of the train instead, the passenger would pass at 29.

In 1632 Galileo asked his readers to imagine being shut in a cabin below the deck of a large ship. While the ship sits in harbour, water drips straight down from a bottle and anything tossed to a friend flies as usual. Now let the ship sail smoothly, at a steady speed in a straight line. Everything in the cabin behaves exactly as before. No experiment done inside can tell you whether the ship is moving.

Steady motion cannot be felt. What your body does feel is a change in velocity: the lurch when the ship speeds up, slows or turns. This is why you sit calmly on a planet that is spinning. At the equator the ground moves east at about 1,670 kilometres an hour, but the motion is so nearly steady that nothing in daily life gives it away.

Throw a ball and it does two things at once. It travels sideways, and it rises and falls. Galileo’s insight was that the two motions do not interfere. With air resistance ignored, nothing pushes the ball sideways once it leaves your hand, so its horizontal speed never changes. Gravity pulls straight down, so it acts only on the vertical part of the motion, which behaves exactly like free fall.

Put the two together, steady travel sideways and an accelerating fall, and the path is a curve called a parabola. Every thrown ball, jet of water and cannonball comes close to tracing one.

The independence has a startling consequence. Fire a bullet level and, at the same instant, drop another from the same height. The fired bullet travels far, yet both hit the ground together, because both fall the same distance under the same gravity.

Two speeds set the timing of the world around you. Sound travels through air at about 340 metres per second, so it needs roughly 3 seconds to cover a kilometre. Light is in another league: about 300,000 kilometres per second, fast enough to circle the Earth more than seven times in one second.

The gap between them turns a thunderstorm into a measuring instrument. Lightning and thunder are made at the same moment. The flash reaches you almost at once, and the thunder follows at the speed of sound. Count the seconds between them, divide by three, and you have the distance to the storm in kilometres.

Even light takes time over great distances. The Sun is about 150 million kilometres away, and its light needs about 8 minutes to arrive. You never see the Sun as it is, only as it was 8 minutes ago.