2.5 Game 2302-0350 units and dimensional analysis

Table of contents

• Preface
  • General
  • Viewing tip
    • Figures
    • Listings
• General background information
  • Formatting mathematical expressions
  • What do we mean by units?
  • SI units
  • Prefixes for powers of ten
  • Dimensional analysis
• Discussion and sample code
  • Manual exercise on converting units
  • Using JavaScript to convert units
  • More substantive JavaScript examples
    • Free fall exercise
    • A plotting exercise
• Run the scripts
• Miscellaneous

Preface

General

This module is part of a series of modules designed for teaching the physics component of GAME2302 Mathematical Applications for Game Development at Austin Community College in Austin, TX. (See GAME 2302-0100: Introduction for the first module in the course along with a description of the course,course resources, homework assignments, etc.)

The purpose of this module is to explain units and dimensional analysis.

Viewing tip

I recommend that you open another copy of this document in a separate browser window and use the following links to easily find and view the Figuresand Listings while you are reading about them.

Figures

• Figure 1 . Single-line format.
• Figure 2 . SI base units.
• Figure 3 . Examples of SI derived units.
• Figure 4 . A sampling of SI prefixes.
• Figure 5 . Screen output for Listing #1.
• Figure 6 . Screen output for Listing #2.
• Figure 7 . Screen output for Listing #3.
• Figure 8 . The trajectory of the falling rock.

Listings

• Listing 1 . Convert from paces to miles.
• Listing 2 . Free fall exercise.
• Listing 3 . A plotting exercise.

General background information

As a young engineering student many years ago, I was told that when evaluating mathematical expressions, I should always embed the units in theexpression, manipulate them algebraically, and confirm that the units that survive until the end of the evaluation are correct relative to the problemspecifications.

Three good reasons

There are at least three good reasons for following this procedure:

• It shows what the units of the result are. A common mistake is to get the correct numerical result of a calculation but to write it with the wrong units.
• It shows where unit conversions must be performed. If units that should have canceled do not cancel, we need to go back and perform the necessary conversions. Forexample, when a speed in miles per hour is being calculated and the result comes out with units of miles per second, we should convert seconds to hours.
• It helps to locate mistakes. If a velocity (meters per second) is being calculated and the units come out as seconds per meter, we know to look for an error.

Formatting mathematical expressions

Multiple-line format

We might typically write a fraction as a horizontal line with the numerator above the line and the denominator below the line using superscripts to indicateexponents. The product of two fractions would typically be written in the same format with a multiplication indicator joining the two fractions.

Single-line format

Figure 1 shows multiplication and division of fractions in a single-line format similar to how we write expressions incomputer programs.