It was meant not only as a joke but also to point out that there is a sense in which an answer of 45 mph is correct. Many students don't realize that the word "average" in the typical expression for "average velocity" (displacement over time) means an average over time. And, since they don't realize this, it makes sense that they would take the average of 30 mph and 60 mph to get 45 mph. Yes, that is an average, but what they don't realize is that it isn't the same type of average used to derive the expression "average velocity = displacement over time" (it is a derivation, after all, and assumes a certain type of average).
There are several examples where an average with respect to distance might be useful. John Denker gave one below. For another, consider that <F> with respect to time is equal to m Delta v/Delta t whereas <F> with respect to distance is equal to Delta K/Delta x (where K is 1/2 mv^2).
Robert A. Cohen, Department of Physics, East Stroudsburg University
570.422.3428 rcohen@esu.edu http://www.esu.edu/~bbq
-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-bounces@carnot.physics.buffalo.edu] On Behalf Of John Denker
Sent: Thursday, September 29, 2011 10:05 PM
To: Forum for Physics Educators
Subject: [Phys-l] distance, rate, time
On 09/29/2011 03:15 PM, Robert Cohen wrote:
Average over time or over space?
I suspect that was meant to be a joke ... but it is even more fun if we take it seriously.
As we all know, you can find the average speed by averaging the instantaneous speed with respect to time:
∫ s dt
〈s〉 = --------- [1]
∫ dt
Now the fun part is that you can find the average inverse speed by averaging the instantaneous inverse speed with respect to *distance*
∫ 1/s dx
〈1/s〉 = ---------- [2]
∫ dx
Equation [1] is useful if you want to find the total distance traveled. That's the numerator in equation [1].
Equation [2] is useful if you want to find the total elapsed time. That's the numerator in equation [2].
Hint: Both numerators are kinda obvious if you think about the dimensions.
The singularity in equation [2] can be taken as a warning. If you are trying to finish a race in the least amount of time, going slowly even for a rather short distance is devastating. This is actually true, and quite noticeable if you do the experiment.
To say the same thing in slightly more positive terms, if there is a fast half of the course and a slow half of the course (perhaps due to hills) you can gain a lot more by speeding up the slow half by 1 mph than you can by speeding up the fast half by 1 mph. Also true and quite noticeable.
====================
This occasionally comes in handy as an answer to the athlete who thinks physics equations could not possibly explain anything he is interested it.
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