Graphics Reference
In-Depth Information
˙
x
(
t
)
x
(
t
)
t
t
1D cannonball position
Velocity
x
(
t
)
x
(
t
)
x
(
t
)
t
t
˙
x
(
t
)
Time‐state path
˙
x
(
t
)
Rolled off roof
Fired from cannon
Weak cannon
Figure 35.27: Cannonball path in time-state space.
the
(
t
,
x
)
-plane are shown to keep the diagram simpler. One thick curve is our
original cannonball, another is a cannonball rolling off a roof, where
x
(
0
)=
0
and
x
(
t
)
0; and the third is a cannonball fired from a weak cannon so that
the initial velocity is small but nonzero. That cannonball strikes the ground early
and has zero velocity and position after collision. There are infinitely many other
solutions for
X
in Equation 35.62, depending on the initial state.
By definition, the initial state of the universe is entirely identified by the value
that we choose for
X
(
0
)
. There is nothing special about
t
=
0. Ignoring forces
from human interaction, for any particular
t
i
the entire future solution for
t
>
t
i
is
determined by
X
(
t
i
)
. In other words, if you give me an
X
(
0
)
, I can tell you
X
(
t
)
for
all subsequent
t
. My response is a curve in
(
t
,
X
)
-space. If you give me a different
X
(
0
)
, I'll give you a different curve. Every point lies in space on one of these
many possible curves. Furthermore, the curves aren't merely geometry: They are
parametric curves, with parameter
t
. That is why it makes sense to compute
X
(
t
)
.
>
Inline Exercise 35.3:
This exercise is mandatory. Don't progress until you've
completed it.
We've just discussed the initial conditions that give rise to three different
(
t
,
X
)
curves shown in the lower-right subfigure of Figure 35.27. Make a copy
by hand of that lower-right subfigure. Think about how we drew the curves and
the equations that govern them—don't just copy the shape.
Now, consider a new scenario. Someone takes the (ill-advised) action of
leaning out a window halfway up the building with a cannon, points it straight
upward, and fires off a cannonball. Write down the equation of motion as a
function
X
(
t
)
and draw the trajectory in time-state space superimposed on the
previous curves.
