Game Engine Architecture

636 12. Collision and Rigid Body Dynamics

means fi nding v(t) and r(t) given knowledge of the net force Fnet(t) and pos-

sibly other information, such as the position and velocity at some previous

time. As we’ll see below, this amounts to solving a pair of ordinary diff er-

ential equations—one to fi nd v(t) given a(t) and the other to fi nd r(t) given

v(t).

12.4.3.1. Force as a Function

A force can be constant, or it can be a function of time as shown above. A force

can also be a function of the position of the body, its velocity, or any number

of other quantities. So in general, the expression for force should really be

writt en as follows:

Fr v()t, ( ), ( ),t t =mt a( ). (12.3) ...

This can be rewritt en in terms of the position vector and its fi rst and second

derivatives as follows:

Fr r()t t,( t ), ( ) , ... = m t^ r( ).

For example, the force exerted by a spring is proportional to how far it has

been stretched away from its natural resting position. In one dimension, with

the spring’s resting position at x = 0, we can write

F t xt(), ()=−kxt (),

where k is the spring constant , a measure of the spring’s stiff ness.

As another example, the damping force exerted by a mechanical viscous

damper (a so-called dashpot) is proportional to the velocity of the damper’s

piston. So in one dimension, we can write

F t vt(), ()=−bvt (),

where b is a viscous damping coeffi cient.

12.4.3.2. Ordinary Differential Equations

In general, an ordinary diff erential equation (ODE) is an equation involving a

function of one independent variable and various derivatives of that function.

If our independent variable is time and our function is x(t), then an ODE is a

relation of the form

dx

dt

f xt dx t

dt

d xt

dt

d xt

dt

n

n

n

= n

⎛

⎝

−

, ( ), −

(), 2 (),, ... ()

2

1

⎜⎜ 1

⎞

⎠

t ⎟.

Put another way, the nth derivative of x(t) is expressed as a function f whose

arguments can be time (t), position (x(t)), and any number of derivatives of

x(t) as long as those derivatives are of lower order than n.