Game Development Reference

In-Depth Information

which is the famous expression of Newton's second law of motion. We will take a closer

look at this equation later.

By no means did we just derive this famous formula. What we did was check its di‐

mensional consistency (albeit in reverse), and all that means is that any formulas you

develop to represent a force acting on a body had better come out to a consistent set of

units in the form (
M
) (
L/T
2
). This may seem trivial at the moment; however, when you

start looking at more complicated formulas for the forces acting on a body, you'll want

to be able to break down these formulas into their component dimensions so you can

check their dimensional consistency. Later we will use actual units, from the SI (
le Sys‐

tème international d'unités
, or International System of Units) for our physical quantities.

Of course, there are other unit systems, but unless you want to show these values to

your gamers, it really does not matter which system you use in your games. Again, what

is important is consistency.

To help clarify this point, consider the formula for the friction drag on a body moving

through a fluid, such as water:

R
f
= 1/2 ρ V
2
S C
f

In this formula,
R
f
represents resistance (a force) due to friction, ρ is the density of water,

V
is the speed of the moving body,
S
is the submerged surface area of the body, and
C
f

is an empirical (experimentally determined) drag coefficient for the body. Now rewriting

this formula in terms of basic dimensions instead of variables will show that the di‐

mensions on the left side of the formula match exactly the dimensions on the right side.

Since
R
f
is a force, its basic dimensions are of the form:

(M) (L/T
2
)

as discussed earlier, which implies that the dimensions of all the terms on the right side

of the equation, when combined, must yield an equivalent form. Considering the basic

units for density, speed, and surface area:

• Density: (M)/(L
3
)

• Speed: (L)/(T)

• Area: (L
2
)

and combining these dimensions for the terms, ρ
V
2
S
, as follows:

[(M)/(L
3
)] [(L)/(T)]
2
[L
2
]

and collecting the dimensions in the numerator and denominator yields the following

form: