Game Development Reference
In-Depth Information
12.1.2 Inertial Reference Frames
If we take the special case where
f
=
0
, then according to Newton's sec-
ond law,
a
=
0
. This is a restatement of his first law. So we see that
if Newton had been just a bit more clever, he could have said the same
thing in only two laws instead of three. Of course, Newton not only broke
through the barrier of “common sense” to create elegant formulas that ex-
plain the workings of every physical system in the entire universe, he also
simultaneously invented a complete branch of the mathematics needed to
fully explore these ideas—calculus. So perhaps he was a clever guy after
all. We assume he had a good reason for keeping his first law; we interpret
it as a statement about reference frames.
The vectors
a
and
f
are specified in some reference frame, and if we
choose a bad reference frame, the equation does not hold. Reference frames
in which the basic mechanical laws hold (especially
f
= m
a
) are known as
inertial reference frames. Coordinate spaces for which this law does not
hold unless we invent fictional forces are called noninertial frames.
For example, imagine a robot eating a herring
sandwich in an elevator. Someone cuts the ele-
vator cables, and the elevator, robot, and sand-
wich begin to fall. Now, this robot has been
programmed with the knowledge that it likes to
eat herring sandwiches,
2
but without any general
sense of self-preservation, so it does not panic. It
looks at the herring sandwich floating in mid-air
instead of falling to the elevator floor, as it would
reasonably expect. The robot, having also been
programmed with an incomplete understanding of
Newton's laws, thinks to itself, “My goodness, this
is quite unusual! I know gravity must be pulling
this sandwich downwards and I know
f
= m
a
, and
since the sandwich is not accelerating downwards,
the net force acting on it must be zero. There-
fore, there must be some upward force acting on
this sandwich. Quite fascinating! What might be
the source of this force? Now, if I calculate. . . ”
CRASH!
Figure 12.1
A robot in a falling elevator
is in a non-inertial frame.
He must invent a fictitious
upward force to counteract
gravity to explain why his
herring sandwich doesn't
fall.
2
You will, of course, recall the herring-sandwich-loving robot from Section 3.3. This
robot is a newer model with improved programming that allows him to pick up the
sandwich without a scoop—a major innovation in the herring-sandwich-eating robot
business.
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