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Could you survive the ride?



The following question came up on a reform calculus mailing list. In its
original form, posed by Ron Sellke, it concerned the difference between a
function whose second derivative was zero at a point, and a function whose
second derivative was undefined at a point. I've taken the liberty of
translating it into a kinematics problem that I think would be of interest
to a calculus-based physics class.

Consider the functions f(t) = t^(7/3) and g(t) = t^(5/3). Let each function
represent the position at time t of a cart moving along a straight track.
Both these functions vanish at the origin, have first derivatives which
vanish at the origin, but have second derivatives which behave much
differently at the origin -- f" vanishes, while g" goes belly up.

The question for the students is to describe what each ride would "feel"
like. They could put their graphing calculators to good use by zooming in
on graphs of these functions and their first and second derivatives near t
= 0. It's surprising how similar the graphs of f and g appear near the
origin, with no indication of how different the second derivatives behave.

Before I throw this out to my class, I was hoping someone on this list
could help me with a question that is bound to come up: Could you survive
the ride in the g-cart? The infinite acceleration as t --> 0 implies an
infinite force which is troublesome. Yet the integral of the acceleration
from 0 to any finite time, which is proportional to the impulse given to
the cart, is finite.

There is also something that bothers me about the f-cart. Even though f" is
zero at t = 0, f''' becomes infinite there. So the force on the f-cart is
zero at t = 0, but it is changing infinitely fast. What would that feel
like?

Paul Goodman
Skyline College
San Bruno, CA