Re: Kinematics of Throwing Balls

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Regarding Jack U.'s comment;

> The answer to this question depends upon your definition of "average".
> David seems to be defining the quantity in a way that makes for maximum
> mystification of the student.  I define "average" as 1/2(V_{f}-V{i}))
> which may not, in general, be the same as the "mean" velocity
> (1/T)integral(_0^T)(v(t)dt).

I don't understand what you are saying here.  If you want to define
the average velocity as an endpoint average then I think you would
want to define it as (1/2)*(V_{f} + V{i}) with a sum of the endpoints
rather than the difference of the endpoints.

But recall that Ludwik's initial formulation of the problem did not
ask about the average velocity at all.  It rather asked about the
average *acceleration*.  Even if one wanted to define the average
acceleration as (1/2)*(a(L) + a(0)) we still have the problem of
figuring out just what a(0) and a(L) actually are given the original
formulation of the problem.  The answer to this is especially
complicated considering that at both x = 0 and x = L the bullet's
acceleration may seem to be undergoing near discontinuities in its
acceleration at precisely those very points.

Maybe you wanted to define the average acceleration as the quotient
of the endpoint difference in velocities divided by the elapsed time,
i.e. (1/T)*(V_{f}-V_{i}).  But in this case it is precisely the
*same* as the integral formulation for the average acceleration, i.e.
(1/T)integral(_0^T)(a(t)dt) by the fundamental theorem of calculus.

In any event Ludwik's reformulation of the problem is currently much
more amenable to treatment by an introductory student.

> These distinctions, IMO, have no place in an
> elementary course where "average" is intended to give the student a sense
> of a relationship of displacement over time.

Certainly the average *acceleration*--no matter how it is defined--
can be intended to give the student "a relationship of displacement
over time".

> See the first chapter of my calculus text which attempts to emphasize this
> concept.

I think we (at least I) may be discussing a somewhat different
concept here.

David Bowman
dbowman@georgtowncollege.edu