How would you answer the question, "Where am I?" You might say you are in a particular classroom. Or perhaps you’re in your bedroom. Regardless of how you describe your position, it will always be relative to something else. Your classroom is a specific room inside of a specific building. You could avoid referencing a building by giving your GPS position, but that position is relative to the earth. There is no such thing as an "absolute" position. Position is always measured relative to something else.

In mathematics, whenever we plot a point on the \(xy\) plane, we always start from a point that’s called the origin. This is the point from which you originate your movement to locate the point. The same concept applies to physics. There will always be some position that is considered the origin for the particular problem, and that’s the point from which all other positions are measured.For example, if we are thinking about a runner in a straight-line race, we might have the starting line be the origin and then measure distances towards the finish line. If we are thinking about the motion of the planets, it might make sense for us to take the sun as the origin.

For kinematics, we are interested in the position of an object at any moment in time. We will start with the one-dimensional problem of the straight-line race runner. When we say that this is a one-dimensional problem, we mean we can describe the position of the runner using just a single number \(x\text{,}\) which in this case is the distance from the starting line to the runner (in meters). Let’s say that the race begins at time \(t = 0\text{.}\) The position of the runner is \(x = 0\) at the start, and it increases over time. Let’s say that at time \(t = 1\) (in seconds) the runner is at \(x = 5\) (meters from the starting line).

If we continue to chart the position of the runner every second, we would get a table of data that gives us an indication of where the runner is at any time. It is important to note that this table only tells us where the runner is at specific instants of time, and we have to try to infer the locations at other times. For example, if the runner is at \(x = 0\) at \(t = 0\) and at \(x = 5\) at \(t = 1\text{,}\) we might guess that the runner is at \(x = 2.5\) at time \(t = 0.5\text{.}\) However, you might also guess that the value of \(x\) might be a little smaller than 2.5 because the runner is still speeding up. While we can make arguments about exactly what that should be, it’s ultimately just an educated guess.

In more complex problems, we might need to track the position of the object using two or three numbers. For example, if we were tracking the location of a car in a city, we might have one number (\(x\)) for the east-west position of the car and another number (\(y\)) for the north-south position. Or if we were tracking the position of a drone, we might need three coordinates because we would also need to keep track of the height (\(z\)).

Technically, position is always a vector. This is because the position is always given by two pieces of information: "How far?" and "In what direction?". When we have just a single variable, it’s implied that one direction is the "positive" direction and the other is negative. When there are many dimensions, the variables together give us the instructions for how to determine the location of the object. The combination of going some distance in one direction followed by another movement is part of why vectors work so well. All of this is naturally communicated in the underlying vector algebra.

One way to represent the position of an object at different moments in time is to use a graph. Typically, the horizontal axis is the time axis and the vertical axis is the position variable (or one of the position variables, if working in multiple dimensions). As noted above, the only times at which we know the position of the object are the times where we have measurements. What happens in between is a guess.

This means that when we plot the graph, the points we have from the chart of data are exact, and if we were to try to sketch a curve that fits those points, we would be doing an interpolation, which means that we are guessing about the shape of the curve. Some students are in the habit of just connecting the dots with straight lines, while others try to smooth out their curve a bit. Both methods are prefectly acceptable and mathematically justifiable. And since we’re not going to be doing a lot of interpolation calculations in this course, we’re not going to get into the details of the mathematics of it all. The key point to keep in mind is that lines are easier to do calculations with but curves give more realistic results.