Mechanics 1: Linear Kinematics and Calculus - 3D Math Primer for Graphics and Game Development - page 486

other hand, if you are solving a problem and you end up with an answer
in pounds per square inch (psi), but you are looking for a value in Pascals,
your answer is probably correct, but just needs to be converted to the
desired units. This sort of reasoning is known as dimensional analysis.
Carrying around the units and treating them as algebraic variables quite
often highlights mistakes caused by different units of measurement, and
also helps make unit conversion a snap.
Because unit conversion is an important skill, let's briefly review it here.
The basic concept is that to convert a measurement from one set of units to
another, we multiply that measurement by a well-chosen fraction that has
a value of 1. Let's take a simple example: how many feet is 14.57 meters?
Looking up the conversion factor, 5 we see that 1 m ≈ 3.28083 ft. This
means that 1 m/3.28083 ft ≈ 1. So let's take our measurement and multiply
it by a special value of “1:”
14.57 m = 14.57 m × 1 ≈ 14.57 m × 3.28083 ft
1 m

≈ 47.80 ft. (11.1)

Our conversion factor tells us that the numerator and denominator of the
fraction in Equation (11.1) are equal: 3.28083 feet is equal to 1 meter.
Because the numerator and denominator are equal, the “value” of this
fraction is 1. (In a physical sense, though, certainly numerically the fraction
doesn't equal 1.) And we know that multiplying anything by 1 does not
change its value. Because we are treating the units as algebraic variables,
the m on the left cancels with the m in the bottom of the fraction.
Of course, applying one simple conversion factor isn't too di cult, but
consider a more complicated example. Let's convert 188 km/hr to ft/s.
This time we need to multiply by “1” several times:
188 km
hr

× 1 hr
3600 s

× 1000 m
1 km

× 3.28083 ft
1 m

≈ 171 ft

s .

11.3 Average Velocity

We begin our study of kinematics by taking a closer look at the simple
concept of speed. How do we measure speed? The most common method
is to measure how much time it takes to travel a fixed distance. For example,
in a race, we say that the fastest runner is the one who finishes the race in
the shortest amount of time.
Consider the fable of the tortoise and the hare. In the story, they decide
to have a race, and the hare, after jumping to an early lead, becomes

5 By “looking it up,” we mean using the Internet. There isn't room in this topic for
tables of information that are easily found online. We needed the space for all of our
opinions, jokes, and useless footnotes.