Re: kinematics, traditional or not

[Tim Folkerts <tfolkert@FHSU.EDU>]

> However, the question (at least as I see it) is about
> motivation.  Why would one want to define v and a?

How can you avoid defining v & a???  To use something you need to have a
common understanding.  You can't even know if F = ma unless you know what
the terms are.


> Why would one want to derive expressions that assume a constant
acceleration?

You don't need to derive the constant acceleration equations.  They
certainly aren't somehow fundamental.  v == dx/dt and a == dv/dt.  That's
it.  That IS kinematics.  You can _describe_ any motion in terms of those
equations.

That said, with these definition you can then explore their ramifications.
WHAT IF the acceleration is constant...  WHAT IF the motion is circular...
WHAT IF acceleration is proportional to distance...  These are just a few
interesting examples of the what the definitions predict.


> It seems to me that one can either:
> (1) examine a case of free-fall empirically, which leads you to define
> acceleration in order to describe the motion, and then use the finding
that
> free-fall is characterized by constant acceleration to derive an
expression
> for x; or
> (2) examine what happens when a force is applied, which leads you to
> Newton's 2nd law, and then use the assumption that gravity is the only
force
> acting during free-fall to predict a constant acceleration and derive an
> expression for x in order to test that prediction.

But how do you "derive an expression for x"?  You define a == dv/dt and v
== dx/dt and integrate.  You still need to derive the exact same equations
for constant x that you thought weren't worth deriving.

------

Now, I agree that too much time is often spent on kinematics and students
often think that x = 1/2at^2 is a universal law.  And there are only so
many problems you can do with speeding cars and falling balls before the
students are bored.

I think that developing kinematics and dynamics in parallel is a great
idea.  Something like ...

"Newton understood the mathematical definitions for velocity &
acceleration:  v = dx/dt and a = dv/dt.  He also recognized the idea of
forces, but he considered a very novel idea, that F = ma.  What are the
implications of these two ideas???
For an object sitting still, F = 0, dv/dt = 0 so F = ma agrees with 0=0
For an object at constant speed ...
For a crate sliding to a stop...
For an object in freefall, ... "

The point is that you can't verify F=ma unless you already have defined a.
But you also don't want to spend 2 months on kinematics with no other goal
in sight.


Tim Folkerts