**The Bouncing Ball**

**Extrapolation...**

One of the simple concepts underlying the concept of acceleration is extrapolation to unattainable limits. This is also one of the concepts Jean Piaget described as being at the edge of human comprehension. We should suspect this as a probable barrier to understanding.

Defining velocity is one of the simplest examples.

Travel at a *constant* velocity of ten feet per second, and you
move twenty feet in two seconds, thirty in three, etc. And five feet
in a half second. (We are talking about velocity rather than speed,
so we must specify a direction: make it north.) When the travel is
at constant velocity dividing the distance traveled by the time it took
always gives the same number, the (constant) speed (include direction,
too, to make it velocity).

When the velocity isn't constant, calculating it isn't so simple. We approximate the value by taking a small interval of time and observe the distance traveled in that short time. We can get as close to the actual value as we wish simply by making the time (and its associated distance) smaller and smaller: the ratio (distance)/(time) doesn't go to zero; it goes to the appropriate value of speed for the instant of time we are closing in upon. This is extrapolation to an unattainable limit. (It's also what the derivative of calculus is all about.)

Of course, since we are examining *velocity* (not speed) the direction
is also important. Here we are tackling another simple concept that
Piaget would consider at the edge of human comprehension: something with
inseverable multiple components. It's yet another barrier to understanding.

Acceleration adds this same additional potential barrier to understanding. Acceleration is to velocity as velocity is to position. Acceleration is the rate of change of velocity. (Velocity is the rate of change of position. Actually, position, like velocity, is a vector, but we have ways of conceptualizing it without facing that complexity. With acceleration, we must face the whole thing.)

When the direction of the velocity changes, we must take the *vector*
difference between two velocities before we divide by the time it took
to make that change. That happens to be the
distance the tip of our velocity gauge (as seen on a nearby page on this
site) moves in that time interval.

For this reason we suggest studying the behavior of velocity and acceleration gauges to gain understanding of acceleration.

So look again at the two possible barriers to understanding of the simple but difficult concept of acceleration:

extrapolation to unattainable limits

conceptualization of inseverable
multiple components