Game Engine Architecture

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a v v

r

r

() () ()

()

( ).

t dt

dt

t

dt

dt

t

==

==





2

2

12.4.2.2. Force and Momentum

A force is defi ned as anything that causes an object with mass to accelerate or
decelerate. A force has both a magnitude and a direction in space, so all forces
are represented by vectors. A force is oft en denoted by the symbol F. When N
forces are applied to a rigid body, their net eff ect on the body’s linear motion
is found by simply adding up the force vectors:

net
1

.

N

i

i=

FF=∑

Newton’s famous Second Law states that force is proportional to accelera-
tion and mass:

Far()tmtmt== () (). (12.2)

As Newton’s law implies, force is measured in units of kilogram-meters per
second squared (kg-m/s^2 ). This unit is also called the Newton.
When we multiply a body’s linear velocity by its mass, the result is a
quantity known as linear momentum. It is customary to denote linear momen-
tum with the symbol p:

pv()t mt= ().

When mass is constant, Equation (12.2) holds true. But if mass is not con-
stant, as would be the case for a rocket whose fuel is being gradually used up
and converted into energy, Equation (12.2) is not exactly correct. The proper
formulation is actually as follows:

() ( ())

() ,

d t dm t

t

dt dt

==

pv

F

which of course reduces to the more familiar F = ma when the mass is constant
and can be brought outside the derivative. Linear momentum is not of much
concern to us. However, the concept of momentum will become relevant when
we discuss angular dynamics.

12.4.3. Solving the Equations of Motion

The central problem in rigid body dynamics is to solve for the motion of
the body, given a set of known forces acting on it. For linear dynamics, this

12.4. Rigid Body Dynamics