Speed, average speed and graphs

Two cyclists reach the same point, but one takes half as long. How can we express the difference? And if a cyclist stopped along the way, could one number describe the whole trip?

Imagine a track marked every 2 m. A cyclist covers exactly one marked section each second. Equal time intervals always give equal distances: this is motion at constant speed. The value 2 m/s tells us how many metres the distance increases by each second. We denote elapsed time by t; its unit is the second, written с in these formulas. Constant speed means an unchanging rate of travel. For the direction to stay unchanged as well, the motion must also be along a straight line.

If we measured a long section, how do we find the distance covered in one second? At constant speed, equal seconds give equal distances, so divide the total distance s by the elapsed time t: v = s/t. Cover twice the distance in the same time and the speed doubles; take twice as long over the same distance and it halves. Using metres and seconds gives m/s. Division requires t > 0: an instant with zero duration is not a measured time interval.

Now suppose the constant speed is known. The distance increases by v metres in one second and by s = vt in t seconds. Double the time without changing the speed and the distance doubles. To find the time needed for a given distance, divide by the distance covered each second: t = s/v. For the same distance, a higher speed means a shorter time; at v = 0, no nonzero distance can be covered. These are three ways of answering different questions about the same constant-speed relationship.

A real walk includes speeding up, walking steadily and stopping. One number cannot recover every detail, but it can let us compare whole trips. Average speed is the constant speed that would cover the same total distance in the same total time. It does not mean the walker maintained that speed throughout. Two walks with the same distance and duration have the same average speed, even if the motion within them is very different.

Why not simply average two speed readings? If we travel fast for a minute but slowly for an hour, the second section has a much larger effect on the result. First add the distances and times: vср = (s₁ + s₂)/(t₁ + t₂), including stops in the denominator. The arithmetic mean of two speeds works only when the two sections take equal times. Make the stop longer without changing the distance: the average speed falls even though the walking speed stays the same.

On an s(t) graph, time is horizontal and cumulative distance is vertical. Take equal time steps and compare how far the line rises: a larger distance increase means a higher speed. A steeper section therefore means faster motion, provided the axis scales are the same. A straight sloping line shows constant speed; a horizontal line shows a stop. Cumulative distance cannot decrease. A line sloping down would have to represent a different quantity, such as coordinate.

Two cars may show the same speed on their speedometers now, while one is braking and the other speeding up. To distinguish them, we need to know how velocity changes over time. Acceleration describes this change. Along a straight line, a change of 2 m/s each second gives an acceleration magnitude of 2 m/s². One ‘second’ belongs to the velocity unit; the other measures how quickly that velocity changes. Turning changes velocity too, because velocity includes direction, even when speed stays constant.

To separate speeding up from the direction of motion, choose a fixed axis. The component vₓ is signed: positive along the axis and negative in the opposite direction. Subtract the initial component from the final one and divide by the elapsed time: aₓ = (vₓ − v₀ₓ)/t. This is the average acceleration component over an interval t > 0. The same change in half the time gives twice the acceleration. If the velocity component stays unchanged, the numerator and the acceleration are zero.

An object moves along the positive axis and slows from 6 to 2 m/s in 2 s. Its velocity component changes by −4 m/s, so aₓ = (2 − 6)/2 = −2 м/с². Here the negative sign accompanies slowing down. Now imagine motion against the axis: the component changes from −2 to −6 m/s. Acceleration is negative again, but speed is increasing! The sign tells us the direction of the velocity change. To decide whether an object speeds up or slows down, compare the directions of velocity and acceleration.

A student walks 300 m in 200 s, then stands still for 100 s. Find the average speed over all 300 s.

The stop adds no distance, so the total is 300 m. Include the stop in the total observation time.

Average speed: 300 м / 300 с = 1 м/с.

This is less than the walking speed, 300/200 = 1,5 м/с, as expected when the journey includes a stop.

The average speed is 1 m/s.

Speed connects distance travelled with elapsed time. Average speed describes the whole journey, including stops.

On a distance–time graph, compare distance increases over equal times. A stop adds no distance; a return journey does.

Acceleration describes a change in velocity, which includes direction. A negative sign alone does not mean slowing down.

We can now describe how velocity changes. Next, we will explore why different objects respond differently to the same kind of action.