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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 constantacceleration?
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:to define
(1) examine a case of free-fall empirically, which leads you
acceleration in order to describe the motion, and then usethe finding
that
free-fall is characterized by constant acceleration to derive anexpression
for x; oris the only
(2) examine what happens when a force is applied, which leads you to
Newton's 2nd law, and then use the assumption that gravity
force
acting during free-fall to predict a constant accelerationand 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.