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One last thing to point out. If we assume the hare learned his lesson and
congratulated the tortoise (after all, let's not attribute to the poor animal
all the negative personality traits!), then at t = t
8
they were standing at
the same place. This means their net displacements from t
0
to t
8
are the
same, and thus they have the same average velocity during this interval.
11.4
Instantaneous Velocity and the Derivative
We've seen how physics defines and measures the average velocity of an
object over an interval, that is, between two time values that differ by some
finite amount ∆t. Often, however, it's useful to be able to speak of an
object's instantaneous velocity, which means the velocity of the object for
one value of t, a single moment in time. You can see that this is not a
trivial question because the familiar methods for measuring velocity, such
as
average velocity =
displacement
elapsed time
=
∆x
∆t
=
x(t
b
) − x(t
a
)
,
t
b
− t
a
don't work when we are considering only a single instant in time. What
are t
a
and t
b
, when we are looking at only one time value? In a single in-
stant, displacement and elapsed time are both zero; so what is the meaning
of the ratio ∆x/∆t? This section introduces a fundamental tool of calcu-
lus known as the derivative. The derivative was invented by Newton to
investigate precisely the kinematics questions we are asking in this chap-
ter. However, its applicability extends to virtually every problem where
one quantity varies as a function of some other quantity. (In the case of
velocity, we are interested in how position varies as a function of time.)
Because of the vast array of problems to which the derivative can be
applied, Newton was not the only one to investigate it. Primitive appli-
cations of integral calculus to compute volumes and such date back to
ancient Egypt. As early as the 5th century, the Greeks were exploring the
building blocks of calculus such as infinitesimals and the method of ex-
haustion. Newton usually shares credit with the German mathematician
Gottfried Leibniz
6
(1646-1716) for inventing calculus in the 17th century,
although Persian and Indian writings contain examples of calculus concepts
being used. Many other thinkers made significant contributions, including
Fermat, Pascal, and Descartes.
7
It's somewhat interesting that many of
6
Ian Parberry is conflicted by this. Although he is British and feels that he should
consequently support Newton's case, his PhD adviser's adviser's . . . adviser back 14
generations ago was Leibniz, and hence he feels he owes him some “familial” loyalty.
7
Pascal and Descartes are PhD adviser “cousins” of Ian Parberry's back to the 16th
generation, but nonetheless he can't help thinking of Monty Python's Philosopher's Song
whenever he thinks of Descartes.
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