Motion and Speed: v = s / t Explained with Animation

Moving compared with what?

Imagine sitting in a train that glides out of the station. The man beside you is not moving at all: he stays in the same place relative to the train. The people on the platform see both of you race past. Neither answer is wrong. Position and motion always need a reference point, the thing you measure from. If the train runs at 20 m/s and you walk along the aisle at 1.5 m/s towards the front, your speed relative to the carriage is 1.5 m/s, and relative to the platform it is 20 + 1.5 = 21.5 m/s. Watch the same passenger in the animation: still to the train, moving to the platform.

Speed is distance over time

Speed answers one question: how far in how long? Divide the distance by the time and you get the speed: v = s / t. Here v is the speed in metres per second, s is the distance in metres and t is the time in seconds. A runner who covers 120 m in 20 s has v = 120 / 20 = 6 m/s. To change to kilometres per hour, multiply by 3.6: 6 × 3.6 = 21.6 km/h. Beside the picture a graph draws itself, with time along the bottom and distance up the side. Constant speed gives a perfectly straight line.

Reading the slope

The steepness of the line is the speed. Pick any two points on it and divide how much the line rises by how much time passed. If it rises 30 m in 5 s, then v = 30 / 5 = 6 m/s, the same as before, wherever you look along the line. That is what 'constant' means: the ratio never changes. Now use the sandbox under the picture. Move the speed slider, press start, and watch the distance, the time and the speed in m/s and km/h grow together.

Two runners, two slopes

Put two runners on the same track. Runner A holds 5 m/s and runner B holds 8 m/s. After 10 s, A has covered 5 × 10 = 50 m and B has covered 8 × 10 = 80 m. On the graph both lines start together, but B's line climbs faster. A steeper line always means a higher speed, so the gap between the runners grows every second; after 10 s it is 30 m. Try it in the sandbox: drag the speed up and the line tilts upward, drag it down and the line leans back towards flat.

A flat line means standing still

What if the runner stops? The distance no longer grows, so the line turns flat. Its slope is zero, and so is the speed: v = 0 / t = 0. A graph is a story you can read: climbing means moving, flat means waiting. Press stop in the sandbox and watch the line level out while the clock keeps running. The distance you have already covered stays on the graph, because stopping does not take it back.

Speeding up bends the line

Real things rarely hold one speed. A car pulling away from a traffic light starts slowly and keeps gaining speed, so its line curves upward. For steady acceleration a from rest, the distance is s = a·t² / 2. With a = 2 m/s², after 5 s the car has gone 2 × 25 / 2 = 25 m, and after 10 s it has gone 2 × 100 / 2 = 100 m. Doubling the time gave four times the distance. That curve is the signature of changing speed, and it returns in the next chapter on falling bodies.

Average and instantaneous speed

Your speedometer shows the instantaneous speed: the speed at this very moment. The average speed is something else, all of the distance divided by all of the time: v_avg = s / t. Suppose a car drives 150 km in 2 h, stopping at lights and slowing in towns. Its average speed is 150 / 2 = 75 km/h, about 20.8 m/s, even though the speedometer showed 100 km/h at some moments and zero at others. On the graph, the average speed is the slope of the straight line joining the start to the end of the trip; the instantaneous speed is the slope of the curve at one point.

Motion and speed