65112.4.7.2. Impulsive Collision Response
When two bodies collide in the real world, a complex set of events takes place.
The bodies compress slightly and then rebound, changing their velocities and
losing energy to sound and heat in the process. Most real-time rigid body
dynamics simulations approximate all of these details with a simple model
based on an analysis of the momenta and kinetic energies of the colliding ob-
jects, called Newton’s law of restitution for instantaneous collisions with no friction.
It makes the following simplifying assumptions about the collision:
z The collision force acts over an infi nitesimally short period of time, turn-
ing it into what we call an idealized impulse. This causes the velocities of
the bodies to change instantaneously as a result of the collision.
z There is no friction at the point of contact between the objects’ surfaces.
This is another way of saying that the impulse acting to separate the
bodies during the collision is normal to both surfaces—there is no tan-
gential component to the collision impulse. (This is just an idealization
of course; we’ll get to friction in Section 12.4.7.5.)
z The nature of the complex submolecular interactions between the bodies
during the collision can be approximated by a single quantity known
as the coeffi cient of restitution , customarily denoted by the symbol ε. This
coeffi cient describes how much energy is lost during the collision. When
ε = 1, the collision is perfectly elastic , and no energy is lost. (Picture two
billiard balls colliding in mid air.) When ε = 0, the collision is perfectly in-
elastic , also known as perfectly plastic , and the kinetic energy of both bod-
ies is lost. The bodies will stick together aft er the collision, continuing to
move in the direction that their mutual center of mass had been moving
before the collision. (Picture pieces of putt y being slammed together.)
All collision analysis is based around the idea that linear momentum is
conserved. So for two bodies 1 and 2, we can write
(^)
12 12
11 2 2 11 2 2
, or
mm mm,
+ =+′′
+ =+′′
pp pp
vv vv^where the primed symbols represent the momenta and velocities aft er the col-
lision. The kinetic energy of the system is conserved as well, but we must ac-
count for the energy lost due to heat and sound by introducing an additional
energy loss term Tlost :
(^) 22 2 211 1 1mv1 1^ 22 2 2+= + +mv^22 mv1 1′′mv^22 Tlost.
If the collision is perfectly elastic, the energy loss Tlost is zero. If it is perfectly
plastic, the energy loss is equal to the original kinetic energy of the system, the
12.4. Rigid Body Dynamics