Game Development Reference
In-Depth Information
Consider this simple example: a 150 gram (0.15 kg) bullet is fired from a gun at a muzzle
velocity of 756 m/s. The bullet takes 0.0008 seconds to travel through the 610 mm (0.610
m) rifle barrel. Calculate the impulse and the average impulsive force exerted on the
bullet. In this example, the bullet's mass is a constant 150 grams and its initial velocity
is 0, thus its initial momentum is 0. Immediately after the gun is fired, the bullet's mo‐
mentum is its mass times the muzzle velocity, which yields a momentum of 113.4 kg-
m/s. The impulse is equal to the change in momentum, and is simply 113.4 kg-m/s. The
average impulse force is equal to the impulse divided by the duration of application of
the force, or in this case:
Average impulse force = (113.4 kg-m/s) / (0.0008 s)
Average impulse force = 141,750 N
This is a simple but important illustration of the concept of impulse, and you'll use the
same principle when dealing with rigid-body impacts. During impacts, the forces of
impact are usually very high and the duration of impact is usually very short. When two
objects collide, each applies an impulse force to the other; these forces are equal in
magnitude but opposite in direction. In the gun example, the impulse applied to the
bullet to set it in motion is also applied in the opposite direction to the gun, giving you
a nice kick in the shoulder. This is simply Newton's third law in action.
In addition to the impulse momentum principle discussed in the previous section, our
classical impact, or collision response, analysis relies on another fundamental principle:
Newton's principle of conservation of momentum, which states that when a system of
rigid bodies collide, momentum is conserved. This means that for bodies of constant
mass, the sum of their masses times their respective velocities before the impact is equal
to the sum of their masses times their respective velocities after the impact:
m 1 v 1- + m 2 v 2- = m 1 v 1+ + m 2 v 2+
Here, m refers to mass, v refers to velocity, subscript 1 refers to body one, subscript 2
refers to body two, subscript - refers to the instant just prior to impact, and subscript
+ refers to the instant just after impact.
A crucial assumption of this method is that during the instant of impact the only force
that matters is the impact force; all other forces are assumed negligible over that very
short duration. Remember this assumption, because in Chapter 10 we'll rely on it when
implementing collision response in an example 2D real-time simulation.
We've already stated that rigid bodies don't change shape during impacts, and you know
from your own experience that real objects do change shape during impacts. What's
happening in real life is that kinetic energy is being converted to strain energy to cause
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