Instantaneous Velocity
The velocity and speed of an object are usually variable. We all
intuitively understand the concept of instantaneous velocity (or
speed). It is what the speedometer reads in a car. Mathematically we
define it the same way as we do average velocity or speed except that
we make $\Delta$t $\rightarrow$ 0 , so it is really how
far the object goes during a very short period of time. The
instantaneous velocity is just called v and, with the aid of
calculus, it is given by
\[ \rm v = \frac{dx}{dt} \]
The instantaneous velocity is the time-derivative of the displacement.
Graphically, v is the slope of the plot of x vs. t. This will be
the next topic.
If the velocity does not change (and only in this case),
then vav = v.
Some notes on the use of
calculus in this course:
- Although most of you have taken calculus, it is
seldom needed to complete the homework assignments
or the exams.
- If you decide that calculus is not for you, that's
fine too. The equivalent formulations without calculus
will usually be available.
- As an easy guide to translation between calculus and
non-calculus concepts, you can make the following
substitution:
- Calculus: dx, dt,
dv, ... => non-calculus Dx,
Dt,
Dv,
...
- Then let Dt
go to the limit 0.
- This just means that in calculus we talk about
infinitely small intervals ("d"),
whereas without the use of calculus we have to use
very, very small, but finite, intervals ("D").
- If you need a little refresher or a primer in
calculus, we have written one for you to aid you in
understanding the concepts.
- If you just need to have a reference to the most
important formulas of calculus, we have included one for
you in the help
appendix.
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In case you want to see how this material is presented without
calculus, here is the equivalent page that
does not use calculus to introduce the concept of instantaneous
velocity.
© MultiMedia
Physics, 1999