Game Engine Architecture

634 12. Collision and Rigid Body Dynamics

become integrals in the limit as the sizes and masses of the litt le pieces ap-

proach zero.)

The center of mass always lies inside a convex body, although it may actu-

ally lie outside the body if it is concave. (For example, where would the center

of mass of the lett er “C” lie?)

12.4.2. Linear Dynamics

For the purposes of linear dynamics , the position of a rigid body can be fully

described by a position vector rCM that extends from the world space origin

to the center of mass of the body, as shown in Figure 12.22. Since we’re using

the MKS system, position is measured in meters (m). For the remainder of

this discussion, we’ll drop the CM subscripts, as it is understood that we are

describing the motion of the body’s center of mass.

y

x

rCM

Figure 12.22. For the purposes of linear dynamics, the position of a rigid body can be fully

described by the position of its center of mass.

12.4.2.1. Linear Velocity and Acceleration

The linear velocity of a rigid body defi nes the speed and direction in which the

body’s CM is moving. It is a vector quantity, typically measured in meters per

second (m/s). Velocity is the fi rst time derivative of position, so we can write

v

r

() r

()

t ( ),

dt

dt

==t

where the dot over the vector r denotes taking the derivative with respect to

time. Diff erentiating a vector is the same as diff erentiating each component

independently, so

vt

dr t

dt

x() x rtx

()

==( ),

and so on for the y- and z-components.

Linear acceleration is the fi rst derivative of linear velocity with respect to

time, or the second derivative of the position of a body’s CM versus time. Accel-

eration is a vector quantity, usually denoted by the symbol a. So we can write