Graphics Reference
In-Depth Information
The distinction between
Δ
x
(
t
)
and
x
(
t
)
is important. We have a rigorous definition
of the acceleration function as a derivative:
x
(
t
+Δ
t
)
−
x
(
t
)
x
(
t
)
−
x
(
t
−
Δ
t
)
x
(
t
) = lim
Δ
t
= lim
Δ
t
.
(35.48)
Δ
t
Δ
t
→
0
→
0
In the presence of an impulse at time
t
0
,
x
(
t
)
is discontinuous at
t
0
. This means
that the lower and upper limits are not equal, so
x
(
t
)
is not differentiable at t
0
.
In other words,
x
(
t
0
)
is not defined (which makes sense, since it is “infinite,” yet
integrates to a finite quantity).
Physicists often employ a Dirac delta “function”
δ
(
t
)
for which
+
∞
δ
(
t
)
dt
=
1 and
(35.49)
−∞
δ
(
t
)=
0
∀
t
=
0
(35.50)
as a notational tool for assigning a value to
x
(
t
0
)
(e.g.,
x
(
t
)=
t
0
)Δ
x
(
t
)
).
δ
(
t
−
However, beware that
(
t
)
is not actually a proper function, and that this notation
conceals the fact that differential and integral calculus cannot represent
x
(
t
)
.
Since the rest of our simulation is based on numerical integration of estimated
derivatives, concealing the facts that
x
is not differentiable at
t
0
and
x
is not inte-
grable at
t
0
is a dangerous practice. We therefore accept that
x
(
t
0
)
does not exist
and define a closely related function that does generally exist:
δ
Δ
x
(
t
)=
x
(
t
)
+
x
(
t
)
−
,
−
(35.51)
where the sign superscripts denote the single-sided limits
x
(
t
)
−
= lim
Δ
t
0
x
(
t
−
Δ
t
)
(35.52)
→
x
(
t
)
−
x
(
t
−
Δ
t
)
= lim
Δ
t
→
0
(35.53)
Δ
t
and
x
(
t
)
+
= lim
Δ
t
0
x
(
t
+Δ
t
)
(35.54)
→
x
(
t
+Δ
t
)
−
x
(
t
)
= lim
Δ
t
→
0
.
(35.55)
Δ
t
In plain language, a plus sign denotes a value immediately after a collision and a
minus sign denotes a value immediately before a collision. The value at the time
of the collision is irrelevant, since we're using the impulse to resolve the collision
and jump from the before value to the after value.
The advantage of moving from velocity to momentum is that the
law of
conservation linear momentum
in physics states that the momentum of a closed
system is always constant. Therefore,
p
i
(
t
)=
i
p
i
(
t
)
.
(35.56)
i
For a system with exactly two objects whose indices are
i
and
j
,thelaw
expands to
p
i
(
t
)+
p
j
(
t
)=
p
i
(
t
)+
p
j
(
t
0
)
,
(35.57)
p
i
(
t
)
p
i
(
t
)=
(
p
j
(
t
)+
p
j
(
t
))
, and
−
−
(35.58)
Δ
p
i
(
t
)=
−
Δ
p
j
(
t
)
,
(35.59)
