Game Development Reference

In-Depth Information

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