# Mechanics

calculus-based physics for scientists and engineers

## Coordinate Systems in 1D

### One-Dimensional Coordinate Systems: Intro

Here I'll introduce coordinate systems with a couple one-sentence paragraphs.

One dimension (1D) is a line.

A 1D coordinate system (1DCS) is an imaginary ruler along that line. The end of the ruler, where the numbers start is called the origin. The direction the numbers increase is the positive axis. The other direction (which it can have, since it is imaginary) is the negative axis. A 1DCS has a single variable (like, say, $$x$$) that represents a particular position along the axis/ruler.

A 1DCS allows us to ascribe a number to a position in space along the line. This allows us to create mathematical, tabular, and graphical representations out of pictorial representations.

A coordinate system is not a graph. You can graph anything. You should visualize a coordinate system superimposed upon your problem like an actual ruler that is taking measurements of the location of objects relative to its origin by ascribing numbers to their positions.

### One-Dimensional Coordinate Systems: Choices

A physical system is real. We don't get to make things up about it. That is good and bad. Our coordinate system is imaginary, which means we can change it, and different people can make different choices about the coordinate system they use to solve a problem. This is also good and bad. The choices we have to make about a 1DCS are:

1. location of origin.
2. direction of position axis.
3. variable name.

Since a coordinate system is a mathematical construct, it does not change or affect the physical system. The physical system will behave the same regardless of the coordinate system you choose to study it. However, these choices change how you describe it mathematically and how it translates into other representations.

Just because the physics of what is happening in the real world is not affected by your coordinate system does not mean any coordinate system is just as good as any other. A clever choice of coordinate system can make a system much easier to solve. A poor choice can make a problem unsolvable mathematically. Making good choices without clear guidelines happens exactly how you expect: through practice and experience.

### Solution

1. A ball starts rolling at one end of a 1-m table and rolls in a straight line. It covers 0.2 m every second for three seconds and then stops.

• Draw a picture of the event.
• Put a coordinate system in your picture with an origin where the object started and pointing in the direction it rolled.
• Write a table that shows what this coordinate system measured for each second through 5 s.
• Draw a graph of position vs. time for your data.
• Repeat the above for a coordinate system with its origin at the location the object stopped and whose positive axis points in the opposite direction.

### Solution

1. Find something you can tip at an angle (like a book). Find something that will slide down the surface (like a coin). Find something that can measure distance (like a ruler, but even a page of lined paper will work where your units of measurement are "lines"). Let it slide slowly enough that you can make rough measurements where it is every second. Using a stopwatch from a phone to get every half-second is even better. (It won't be that accurate, but who cares.) Make a table of your data and a graph.