Game Development Reference
In-Depth Information
trajectory model. Simple closed-form solutions to the equations of motion are no longer possible,
but the equations can be solved using an ODE solver.
Some of the key points of the projectile trajectory model with drag effects are as follows:
•
Drag force acts in the opposite direction to the velocity. The magnitude of the drag force
is proportional to the square of the velocity.
•
The three components of motion are coupled when drag is taken into account. The
x-component of drag force will depend on the y- and z-velocity components and so on.
The x-, y-, and z-components of velocity and position can no longer be computed
separately.
•
The drag force is a function of the projectile geometry and is proportional to both the
frontal area and drag coefficient of the projectile.
•
The acceleration due to drag is inversely proportional to the mass of the projectile. Other
things being equal, a heavier projectile will show fewer drag effects than a lighter
projectile.
•
The drag on an object is proportional to the density of the fluid in which it is traveling.
Wind Effects
When we developed the aerodynamic drag model previously in this chapter, there was an implicit
assumption that the projectile was traveling through still air. Now let's turn our attention to
modeling the effect of wind on a projectile trajectory. There are a lot of instances when wind
will affect the trajectory of a projectile. Sharpshooters must take into account wind when they
aim their rifles towards the target. Golfers must similarly change the aim of their shots if there
is a crosswind. Baseball players have a much easier time hitting a home run if the wind is
blowing out than if it is blowing in.
When wind is incorporated into our projectile trajectory model, some simplifications will
be made. The velocity components of the wind will be assumed to be in the x-y plane only—in
other words, it is assumed that there is no vertical component to the wind. The x- and y-direction
wind velocity components will be designated as
v
wx
and
v
wy
.
To understand the effect of wind on a projectile, consider this question: what is the differ-
ence in the drag force experienced by an object traveling at 10
m/s
through still air and a similar
object at rest experiencing a 10
m/s
headwind? The answer is that the two drag force values will
be the same, because the
apparent velocity
seen by the two objects is the same.
■
Tidbit
The concept of apparent velocity is used to test new airplane concepts. A scale model of the plane
is built and placed into a facility known as a
wind tunnel
. The model is fixed in place and large fans blow air
over the stationary model to evaluate how the airplane would perform in flight. The largest wind tunnel in the
United States is at the NASA Ames Research Center in Moffett Field, California. The test section is 80 feet wide
and 40 feet high, and can test full-sized airplanes.
