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Re: [ap-calculus] Average Velocity
Posted:
Oct 28, 2009 10:00 AM
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The student is right! The average velocity across an interval is always
equal to the mean of the instantaneous velocity.
This statement depends upon a careful distinction between the average
velocity and the mean velocity.
The average velocity is defined on the position vs time graph. Across a
given time interval the average velocity is the constant velocity which
corresponds to the same displacement in the same time interval as the given
velocity graph. This is just the slope of the secant line.
The mean velocity is defined on the velocity vs time graph. Given a
velocity graph and a time interval the mean velocity corresponds to the height of
a horizontal (velocity) line which has the same area across the interval
as the area between the velocity graph and the time axis across the given
interval. (The better definition is: the net area between the velocity graph
and the mean graph across the interval is zero. This definition makes the
connection between the discrete and continuous cases. See the comment below.)
Since the integral of the velocity across the time interval is equal to the
mean value of the velocity times the length of the interval, the
Fundamental Theorem of Calculus implies that the mean value of the velocity on the
velocity vs time graph is equal to the average value of the velocity on the
position vs time graph.
This student's conjecture is equivalent to the Fundamental Theorem of
Calculus. Assuming either one implies the other.
It is very important.
Bill
PS. This distinction is (almost) trivial in the case of discrete data but
is still an important distinction to make. The average is defined as the
constant distribution having the same total for the same number of data
points. The height of this distribution is just the sum of the data values
divided by the number of data points. The mean is the location in the data set
with zero total deviation. A short calculation involving
Sigma ( Xi - M ) = 0
shows that the mean is equal to the average.
In a message dated 10/27/2009 4:53:43 P.M. Eastern Daylight Time,
jfunk46360@zenbe.com writes:
Hello all,
I need some clarification on a student's question that I could not answer
today. We know that the average velocity of a particle is the slope of the
secant line connecting two endpoints on the graph of the position
function. The student pointed out that the same result could be found by finding
the mean of the instantaneous velocities at the same two points. Is this
something that I should know, and is it important?
Thanks,
Jim Funk
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