Game Development Reference

In-Depth Information

Figure 2-10. Linear velocity due to angular velocity

Differentiating the equation,
v
=
r
ω:

dv/dt = r dω/dt

yields this formula for the tangential linear acceleration as a function of angular accel‐

eration:

a
t
= r α

Note that there is another component of acceleration for the particle that results from

the rotation of the rigid body. This component—the
centripetal
acceleration—is normal,

or perpendicular, to the circular path of the particle and is always directed toward the

axis of rotation. Remember that velocity is a vector and since acceleration is the rate of

change in the velocity vector, there are two ways that acceleration can be produced. One

way is by a change in the magnitude of the velocity vector—that is, a change in speed

—and the other way is a change in the direction of the velocity vector. The change in

speed gives rise to the tangential acceleration component, while the direction change

gives rise to the centripetal acceleration component. The resultant acceleration vector

is the vector sum of the tangential and centripetal accelerations (see
Figure 2-11
). Cen‐

tripetal acceleration is what you feel when you take your car around a tight curve even

though your speed is constant.