Game Development Reference
In-Depth Information
Here we calculate the moment due to the impulse by taking the vector cross product of
the impulse with the distance from the body's center of gravity to the point of application
of the impulse.
By combining all of these equations with the equation for e and following the same
procedure used when deriving the linear impulse formula, you'll end up with a formula
for | J | that takes into account both linear and angular effects, which you can then use
to find the linear and angular velocities of each body immediately after impact. Here's
the result:
| J | = -( v r n )(e + 1)/[1/m 1 + 1/m 2 + n •(( r 1 × n )/ I 1 ) × r 1 + n
(( r 2 × n )/ I 2 ) × r 2 ]
Here v r is the relative velocity along the line of action at the impact point P , and n is a
unit vector along the line of action at the impact point pointing out from body 1.
With this new formula for | J |, you can calculate the change in linear and angular ve‐
locities of the objects involved in the collision using these formulas:
v 1+ = v 1− + (| J | n )/m 1
v 2+ = v 2− + (−| J | n )/m 2
ω 1+ = ω 1− + ( r 1 × | J | n )/ I 1
ω 2+ = ω 2− + ( r 2 × −| J | n )/ I 2
As we said earlier, we'll show you how to implement these formulas for impulse in code
when you get to Chapter 10 .
Friction acts between contacting surfaces to resist motion. When objects collide in any
type of collision except direct impact, for that very brief moment of contact, they will
experience a friction force that acts tangentially to the contacting surfaces. Not only will
this tangential force change the linear velocities of the colliding objects in the tangential
direction, but it will also create a moment (torque) on the objects, which tends to change
their angular velocities. This tangential impulse combined with the normal impulse
results in an effective line of action of the total collision impulse that is no longer per‐
pendicular to the contacting surfaces.
In practice, it is very difficult to quantify this collision friction force due to the fact that
the friction force is not necessarily constant if the collision is such that the friction force
does not develop beyond the maximum static friction force. Further complications stem
from the fact that objects do tend to deform when they collide, creating an additional
source of resistance. That said, since the friction force is a function of the normal force
between the contacting surfaces, you know that the ratio of the normal force to the
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