Game Development Reference

In-Depth Information

the objects to deform. (See the sidebar
“Kinetic Energy” on page 106
for further details on

this topic.) When the deformation in the objects is permanent, energy is lost and thus

kinetic energy is not conserved.

Kinetic Energy

Kinetic energy is a form of energy associated with moving bodies. It is equal to the energy

required to accelerate the body from rest, which is also equal to the energy required to

bring the moving body to a stop. As you might expect, kinetic energy is a function of

the body's speed, or velocity, in addition to its mass. The formula for linear kinetic energy

is:

KE
linear
= (1/2) m v
2

Angular, or rotational, kinetic energy is a function of the body's inertia and angular

velocity:

KE
angular
= (1/2) I ω
2

Conservation of kinetic energy between two colliding bodies means that the sum of

kinetic energy of both bodies prior to impact is equal to the sum of the kinetic energy

of both bodies after impact:

m
1
v
2
1-
+ m
2
v
2
2-
= m
1
v
2
1+
+ m
2
v
2
2+

Collisions that involve losses in kinetic energy are said to be
inelastic
, or
plastic
, colli‐

sions. For example, if you throw two clay balls against each other, their kinetic energy

is converted to permanent strain energy in deforming the clay balls, and their collision

response—that is, their motion after impact—is less than spectacular. If the collision is

perfectly inelastic
, then the two balls of clay will stick to each other and move together

at the same velocity after impact. Collisions where kinetic energy is conserved are called

perfectly elastic
. In these collisions, the sum of kinetic energy of all objects before the

impact is equal to the sum of kinetic energy of all objects after the impact. A good

example of elastic impact (though not
perfectly
elastic) is the collision between two

billiard balls where the ball deformation is negligible and certainly not permanent under

normal circumstances.

Of course, in reality, impacts are somewhere between perfectly elastic and perfectly

inelastic. This means that for rigid bodies, which don't change shape at all, we'll have to

rely on an empirical relation to quantify the degree of elasticity of the impact(s) that

we're trying to simulate. The relation that we'll use is the ratio of the relative separation

velocity to the relative approach velocity of the colliding objects: