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.
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, firstname.lastname@example.org writes:
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?