# Distance and Displacement

If someone tells you they live 200km from you, do you know where they are? No: they could be anywhere on a circle with radius 200km. To know where they are, you also need to know which direction they are. They need to tell you they are 200km south. This difference between '200km' and '200km south' is the difference between distance and displacement.

Displacement tells you both the distance and direction.

Distance is the length of a line between two points, and has no direction component.

## Displacement can be positive or negative

Fig. 1: Total displacement of C relative to A = displacement C – displacement A = +2m -(- 2m) = +4m. Note that the displacement of A relative to C would be -4m.

Displacement is a vector. That means it can be drawn like an arrow, with a magnitude and a direction.

In the figure, we see three clones on a line. They are each two metres apart. How can we describe their positions more exactly? Well, to do this we need to select a reference point to call zero, and describe the others relative to that point.

In the diagram Clone B is at zero, and Clone C has a displacement of +2m. Clone A has a displacement of -2m. The minus sign indicates that he is in the opposite direction to Clone C and 2m distant from zero.

There is no good reason that positive is to the right, and negative to the left - just convention. You could also create a system with positive going up and negative down, and vice-versa. We will use that system when we push our poor clones out of an aeroplane to study gravity.

Figure 1 is like a photograph - one moment in time. It does not tell us about how their position is changing in time, like a video would. To represent movement in time, we can draw a graph of displacement over time:

In the above graph, each of the lines is horizontal. This means the position is not changing over time, so we can say the person or object is at rest.

In the next graph, the displacement (position relative to zero) is changing over time:

Graph of two objects moving at different speeds.

The blue line is steeper (has a greater incline) than the red line. This indicates that the object described by the blue line is moving faster than the red line object. The slope of the displacement-time curve is therefore a measure of the object's velocity.

## Moving in a circle

When you run around an athletics track, what is the distance and your displacement?

Each time you come around to the starting point, your displacement returns to zero. But the distance you have covered is 2πr, where r is the radius of the circle. the distance you have travelled is always increasing.

Your maximum displacement is 2r, when you are on the opposite side of the track. Then it returns to zero when yo complete a full lap.

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