Game Engine Architecture

652 12. Collision and Rigid Body Dynamics

primed kinetic energy sum becomes zero, and the bodies stick together aft er

the collision.

To resolve a collision using Newton’s law of restitution, we apply an ide-

alized impulse to the two bodies. An impulse is like a force that acts over an in-

fi nitesimally short period of time and thereby causes an instantaneous change

in the velocity of the body to which it is applied. We could denote an impulse

with the symbol ∆p, since it is a change in momentum (∆p = m∆v). However,

most physics texts use the symbol pˆ (pronounced “p-hat”) instead, so we’ll

do the same.

Because we assume that there is no friction involved in the collision, the

impulse vector must be normal to both surfaces at the point of contact. In oth-

er words, pnˆˆ=p , where n is the unit vector normal to both surfaces. This is

illustrated in Figure 12.27. If we assume that the surface normal points toward

body 1, then body 1 experiences an impulse of pˆ, and body 2 experiences an

equal but opposite impulse. Hence, the momenta of the two bodies aft er the

collision can be writt en in terms of their momenta prior to the collision and

the impulse pˆ as follows:

p pp 11 ′′=+ˆˆ; p p p2 2= −;

(^) mm 11 v′′=+ 11 vpˆˆ; m m 22 v= − 22 v p;
(12.9)
The coeffi cient of restitution provides the key relationship between the rela-
tive velocities of the bodies before and aft er the collision. Given that the cen-
ters of mass of the bodies have velocities v 1 and v 2 before the collision and
v′ 1 and v′ 2 aft erward, the coeffi cient of restitution ε is defi ned as follows:
()().vv′′ 21 − =εvv 21 − (12.10)
Solving Equations (12.9) and (12.10) under the temporary assumption
that the bodies cannot rotate yields
n
p Body 2 Body 1
^
Figure 12.27. In a frictionless collision, the impulse acts along a line normal to both surfaces
at the point of contact. This line is defi ned by the unit normal vector n.
11 22
12

ˆˆ

pp;.

mm

vv′′=+n vv= −n