Isn't the confusion about whether the integral of v(t) is position or
displacement really cause by confusing definite integrals and indefinite
integrals?
The inverse of taking the derivative of a function is the indefinite
integral. It is the indefinite integral of velocity that gives us
position (as a function of time) plus a constant.
The definite integral, once the endpoints are defined, is not a
function, but a number. It is the definite integral that is the "area
under the curve." The definite integral of v(t) is displacement. This
displacement can be viewed as a function of time if we view the limits
of integration as functions of time.
If I have position (relative to some reference point as Denker says)
depicted as x(t), then the derivative of this is v(t), which is not a
number, but another function of time. If I take the indefinite integral
of v(t) then I get back the function x(t) except I don't get back x(t=0)
because that was lost when I took the derivative of x(t) to get v(t).
On the other hand, if I take the definite integral of v(t), then I get a
number, not a function, and this number is delta-x. This number is not
determined until I specify the limits of integration.
When I teach this I try to make this distinction between numbers and
functions, and between indefinite integrals and definite integrals.
This is made difficult by textbooks that tend to leave out the explicit
functional notation (for example they write x instead of x(t)), and they
also leave out the explicit notation for change (for example they write
x instead of delta-x). Thus, when the student sees "x" in the text,
does it mean delta-x or does it mean x(t)?
It is cumbersome, but in my board work I try to remember to write
delta-x and x(t) as necessary, that is, I avoid writing only x. I do
the same thing with delta-v and v(t).
I write, for example...
x(t) = integral(v(t)) + constant
delta-x = definiteintegral(v(t)) with limits t1 and t2
delta-x = x(t2) - x(t1)
Michael D. Edmiston, Ph.D.
Professor of Physics and Chemistry
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu