Game Development Reference
In-Depth Information
The trick in applying this formula is in determining the right value for the drag coeffi‐
cient. Just for fun, let's assume a drag coefficient of 0.5 and calculate the terminal velocity
for several different objects. This exercise will allow you to see the influence of the
object's size on terminal velocity. Table 6-5 gives the terminal velocities for various
objects in free fall using an air density of 1.225 kg/m 3 (air at standard atmospheric
pressure at 15°C). Using this equation with density in kg/m 3 means that m must be in
kg, g in m/s 2 , and A in m 2 in order to get the terminal speed in m/s. We went ahead and
converted from m/s to kilometers per hour (km/h) to present the results in Table 6-5 .
The weight of each object shown in this table is simply its mass, m , times g .
Table 6-5. Terminal velocities for various objects
Weight (N)
Area (m 2 )
Terminal velocity (km/h)
Skydiver in free fall
Skydiver with open parachute
Baseball (2.88 in diameter)
4.19×10 −3
Golf ball (1.65 in diameter)
1.40×10 −3
Raindrop (0.16 in diameter)
3.34×10 −4
1.29×10 −5
Although we've talked mostly about spheres in this section, the discussions on fluid flow
generally apply to any object moving through a fluid. Of course, the more complex the
object's geometry, the harder it is to analyze the drag forces on it. Other factors such as
surface condition, and whether or not the object is at the interface between two fluids
(such as a ship in the ocean) further complicate the analysis. In practice, scale model
tests are particularly useful. In the Bibliography , we give several sources where you can
find more practical drag data for objects other than spheres.
Magnus Effect
The Magnus effect (also known as the Robbins effect ) is quite an interesting phenom‐
enon. You know from the previous section that an object moving through a fluid en‐
counters drag. What would happen if that object were now spinning as it moved through
the fluid? For example, consider the sphere that we talked about earlier and assume that
while moving through a fluid such as air or water, it spins about an axis passing through
its center of mass. What happens when the sphere spins is the interesting part—it ac‐
tually generates lift! That's right, lift . From everyday experience, most people usually
associate lift with a wing-like shape such as an airplane wing or a hydrofoil. It is far less
well known that cylinders and spheres can produce lift as well—that is, as long as they
are spinning. We'll use the moving sphere to explain what's happening here.
From the previous section on drag, you know that for a fast-moving sphere there will
be some point on the sphere where the flow separates, creating a turbulent wake behind
the sphere. Recall that the pressure acting on the sphere within this turbulent wake is
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