 Now, location and space requires two or three coordinates, which we can study using vectors. But to simplify the maths a bit, we're going to look at the case when location needs just one coordinate. Once you're comfortable doing the maths with one coordinate, you'll be able to move on to working with more dimensions. And in fact, a single coordinate is enough to keep track of speed along a fixed path, such as a car on the highway, a train on its tracks, or an airplane on a straight flight path. So to describe where a train is along a track, we just have to write down where the centre of mass is at each point in time, x as a function of t. There are a few decisions to make first. We must decide where the zero point along the track is, and maybe also what will be the zero point in time. And then we have to decide which direction along the track is going to be positive. It doesn't matter too much how you choose these things, you just have to be consistent for the whole time you're working on a particular problem. Imagine I record the position of my train as a function of time, and when I plot this information it looks like a straight line. Let's suppose the time axis tick marks are one second each, and the x-axis tick marks are 10 meters. Then we can calculate how fast the train is going as delta x over delta t, and for example we get six metres per second. I can choose a longer time interval, say three seconds, and see that the displacement is 18 metres, so the velocity is still six metres per second. In fact any time interval I choose will give me velocity equals six metres per second. The velocity is constant, and a graph of the velocity as a function of time is just a horizontal line. Now if we think about acceleration, there's no change in velocity, and so the acceleration is zero. Now let's look at a graph of x that is not a straight line. We see from this graph that the train travels 10 kilometres in a 10 minute interval, and we can calculate the average velocity for this trip as 60 kilometres per hour. This is the slope of the line that joins the point at x for time equals zero to x at time equals 10 minutes. If the train had had a constant velocity for those 10 minutes, the x of t curve would look like that red line. Now what further information does the blue curve tell us? If we take a much smaller time interval at the beginning here, there is no change in position, so the velocity is zero. And if we do the same at a later time, we see a steeper slope. These slopes calculated for very small time intervals are called the instantaneous velocities. The lines that just touch the curve here are called tangents to the curve, and this is something you'll study more in calculus.