Game Development Reference
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Figure 4-4. Acceleration versus time
You'll notice that the ship's speed approaches the steady state speed of 20 speed units,
assuming that the propeller thrust remains constant. This corresponds to a reduction
in acceleration from a maximum acceleration at time 0 to no acceleration once the steady
speed is achieved.
This example illustrates how to set up the differential equations of motion and integrate
them to find velocity, displacement, and acceleration. In this case, you were able to find
a closed-form solution—that is, you were able to integrate the equations symbolically
to derive new ones. You could do this because we imposed enough constraints on the
problem to make it manageable. But you can readily see that if there were more forces
acting on the ship, or if the thrust were not held constant but was some function of
speed, or if the resistance were a function of speed squared, and so on, the problem gets
increasingly complicated—making a closed-form solution much more difficult, if not
impossible.
Particle Kinetics in 3D
As in kinematics, extending the equations of motion for a particle to three dimensions
is easy to do. You simply need to add one more component and will end up with three
equations as follows:
ΣF x = m a x
ΣF y = m a y
 
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