Game Development Reference
ΣF z = m a z
The resultant force and acceleration vectors are now:
a = a x i + a y j + a z k
a = a x 2 + a y 2 + a z 2
Σ F = ΣF x i + ΣF y j + ΣF z k
ΣF = ( ΣF x ) 2 + ( ΣF y ) 2 + ( ΣF z ) 2
To hammer these concepts home, we want to present another example.
Let's go back to the cannon example program discussed in Chapter 2 . In that example,
we made some simplifying assumptions so we could focus on the kinematics of the
problem without complicating it too much. One of the more significant assumptions
we made was that there was no drag acting on the projectile as it flew through the air.
Physically, this would be valid only if the projectile were moving through a vacuum,
which, of course, is unlikely here on Earth. Another significant assumption we made
was that there was no wind to act on the projectile and affect its course. These two
considerations, drag and wind, are important in real-life projectile problems, so to make
this example a little more interesting—and more challenging to the user if this were an
actual game—we'll add these two considerations now.
First, assume that the projectile is a sphere and the drag force acting on it as it flies
through the air is a function of some drag coefficient and the speed of the projectile.
This drag force can be written as follows:
F d = -C d v
F d = -C d v x i - C d v y j - C d v z k
where C d is the drag coefficient, v is the velocity of the projectile ( v x , v y , and v z are its
components), and the minus sign means that this drag force opposes the projectile's
motion. Actually, we're cheating a bit here since in reality the fluid dynamic drag would
be more a function of speed squared. We're doing this here to facilitate a closed-form
solution. Also, the drag coefficient here would be determined experimentally for each
shape. Later, we'll discuss how experimental data is used on basic shapes to give data for
Second, assume that the projectile is subjected to a blowing wind, the force of which is
a function of some drag coefficient and the wind speed. This force can be written as
F w = -C w v w
F w = -C w v wx i - C w v wz k