Background Information and Techniques - Computer Animation: Algorithms and Techniques - page 470

B.7 Physics primer
Physically based motion is a limited simulation of physical reality. This can be as simple or as complex
as the implementation requires. In the following sections are some of the equations of importance in
simple physics simulation that can be found in any one of several standard texts. The Mechanical
Universe [ 5 ] is used as the source for the brief discussion that follows.

B.7.1 Position, velocity, and acceleration
The fundamental equation relating position, distance, and speed is shown in Equation B.90 . This can be
used to control the positioning of an object for any particular frame of the animation because the frame
number is tied directly to time ( Eq. B.91 ). The average velocity of a body is the distance moved divided
by the time it took to move, as stated by Equation B.92 . Notice that the unit of velocity is distance per
time, for example, feet/second.

distance ¼ speed time (B.90)

time ¼ frameNumber timePerFrame (B.91)

¼ =

time (B.92)
For this discussion, distance as a function of time is s ( t ); the average velocity from time t 1to t 2is
( s ( t 2)

average Velocity distanceTraveled

t 1). The instantaneous velocity is determined by moving t 2 closer and closer to t 1.
In the limiting case this becomes the derivative of the distance function with respect to time. Similarly,
the average acceleration of an object is the change in velocity divided by the time it took to effect the
change. This is presented in Equation B.93 , where v ( t ) is a function that gives the velocity of the object
at time t. Notice that the unit of acceleration is velocity per time or distance per time, for example, feet/
second 2 . In the same way, instantaneous acceleration is the derivative of v ( t ) with respect to time
( Eq. B.94 ) . In the case of motion due to gravity, g is the acceleration due to gravity—a constant that
has been measured to be 32 feet/second 2 or 9.8 meters/second 2 ( Eq. B.95 ) .

s ( t 1))/( t 2

a ave ¼ vtðÞvtðÞ ð Þ= t 2 t 1 ð Þ (B.93)

0 00

aðÞ¼v ðÞ¼s ðÞ (B.94)

aðtÞ¼g
vðÞ¼gt
sðÞ¼

(B.95)

2

ð gt 1 = 2

B.7.2 Circular motion
Circular motion is important in physics and arises for a variety of phenomena, including the movement
of planets and robotic armatures. Circular motion is easily specified by using polar coordinates. The
position of a particle orbiting the origin at a distance r can be described using Equation B.96 . Here, i and
j are orthonormal unit vectors (at right angles to each other and unit length) and p ( t ) is the positional
vector of the particle. In a constant radius circular orbit, y ( t ) varies as a function of time, and the dis-
tance r is constant. During uniform circular motion, y ( t ) changes at a constant rate, and angular velocity