When I ask students to calculate the magnitude of the average velocity for a
simple projectile problem (e.g., ball thrown horizontally from a roof), they
often simply add the initial and final speeds and divide by 2. They no
doubt think, "when acceleration is constant (as it is in projectile motion),
this is how you can calculate average velocity (in one dimension); and speed
is the magnitude of velocity, right?"
Does anyone have a good brief intuitive explanation (for beginning
calculus-based-physics students) of the reason(s) that this doesn't work?
(or maybe a reference to a particularly good textbook explanation)
I suppose this is a two-part problem: 1) Why is the average of the
magnitude of a vector not the same as the magnitude of the average? 2) Why
is the quantity they are calculating not even the correct average speed?
[Since this is somewhat related to the current discussion entitled
"Problem", which I haven't been following in complete detail, I hope this
request doesn't duplicate something that was just covered there.]