Further Reading

Location in space -- Coordinates


While you've all used graphs and coordinate systems in your math classes, to describe motion we have to take the additional step of tying the coordinate system to the physical world. The two-axis graph in math (an "x-y plot") is a mathematical structure that allows us to use the tools of both geometry and algebra.

But in tying a graph to the physical world, we are doing more. We are making a mathematical model of something in the physical world — something  non-trivial, both in making sense of the physics concepts and in deciding how much of the math we get to legitimately use. Let's go through the process carefully.

We've talked about how we can assign a number to a length using an operational definition by comparing a standard to the length we want to quantify. This isn't good enough for describing location. Consider the following story.

Where was he?

A fisherman went out in the early morning along the river to fish. At the first spot the fish weren't biting at all, so after about half an hour he moved to another spot. Still nothing. So he moved again. The sun was getting high now, so he was getting concerned that he had missed the best time of day. But at his third spot, the fish were biting like crazy and he pulled in a satisfactory haul. When he was done, he wanted to remember this spot. Fortunately, he had remembered to bring a can of paint and a brush, so he painted an "X" on the bottom of his boat.

This is clearly silly.  If he paints the "X" on his boat, it moves with him and it won't help him find the place on the next day. To be able to find it again he needs a marker that is a fixed reference that he can use as a starting point to find the places he wants to find. He needs:

  • a starting point,
  • a direction to go in, and
  • a distance to go along that direction. 

These are what we need to set up a specification of position that communicates where something is. We call the way we do it a spatial coordinate system.

Creating a spatial coordinate system

A spatial coordinate system is a very particular kind of graph; it is one in which the points on the graph are meant to correspond to the points in real space — like a map. 

In general to specify a position, since we live in three dimensional (3D) space, we will need 3 numbers. For example, if I want to tell you to meet me at my favorite Chinese restaurant in Washington, DC, I can tell you it's at the corner or 7th street and H street NW on the 3rd floor. 7th street gives you an east-west location, H street gives you a north-south location, and 3rd floor tells you the vertical location. (And "NW" tells you which quadrant of the city since DC doesn't choose to use negative numbers like we do.)

But for most of the examples we'll deal with in this class, we'll restrict our motions to one or two dimensions (1D or 2D) so we can use a plane. 

Even in 2D there are three independent steps to creating a coordinate system tied to a physical space.

  1. Choose a reference point (origin).
  2. Choose two axes (called here x and y and taken to be perpendicular to each other)
  3. Choose a length scale for measuring distances (here taken to be the same in both directions - the "m" on the graph stands for "meters").

In a spatial coordinate system, a curve might represent a path an object follows. Since an object can go anywhere, the curve can go back and forth, cross itself, and do lots of other things that graphs in a math class don't usually do. (In math the term coordinate system by itself is often used to represent the axes on any kind of graph and we will also do that.)

(Most of the graphs we will draw in this class won't be like this but will be more abstract and need interpretation. See Kinematic graphs.)

Conventions for spatial coordinate systems

There are a number of conventions that we will apply in this class for creating spatial coordinate systems. 

  • The two axes cross at the origin.
    • Sometimes in non-spatial coordinate systems the origin is not shown.  This is called a suppressed zero and might be used to magnify the variation in a curve. (But it is often done for the purpose of misleading the viewer into thinking an effect is more important than it really is.)
  • The positive direction of the axis is indicated with an arrowhead.
    • The other direction is negative.
  • The axes are labeled including specifying the unit in which the axis is measured.
    • Because we are mapping something physical, as always, units are crucial.

These conventions will turn out to be really important since we will be making many different kinds of graphs and things can get very confusing when they are not followed. 

Location vs displacement and length

Three concepts associated with the measurement of location are often confused: position, displacement, and length. If we are specifying something's location by giving its position along a line, we might give its coordinate, $x$. If an object moves from point $x_1$ to a point $x_2$ it has moved a distance, $\Delta x = x_2 - x_1$, a displacement. If we are specifying the size of an object we might write that it's length is the difference of the positions of its endpoints: $L = x_2 - x_1$. All three, $x$, $\Delta x$, and $L$, have dimensions [$x$] = [$\Delta x$] ={$L$] = L and all are measured in the same units. But they mean very different things. This can be very confusing, since equations describing motion given in high school physics classes often write position, $x$ when what is really meant is displacement, $\Delta x$. While you can get away with this if you always start your displacements at the position $x = 0$ (and similarly for choosing your start time at $t = 0$) but our problems will not be that simple and we will not be able to get away with this. You will have to be careful to separate these items conceptually. (See the page Values, change, and rates of change.)

Joe Redish and Wolfgang Losert 9/2/12


Article 308
Last Modified: April 9, 2019