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Re: kinetic energy paradox?



At 02:57 PM 10/22/01 -0400, Joe Heafner wrote:
My quandry is this: why is Earth's kinetic energy conserved and the ball
bearing's kinetic energy NOT conserved? In both cases, an impulse is
imparted perpendicular to the velocity.

At 09:32 AM 10/23/01 -0400, Carl E. Mungan wrote:

It is amusing to push this idea to its limits and consider earth
moving around the sun in an equilateral triangle (with very slightly
rounded corners).

Then earth would go around the sun in a nearly equilateral figure. ...
Yet its KE would still be constant.

This is amusing, but it doesn't answer Joe's question. In fact it
highlights the core of the question. The gravity impulses for the
polygonal orbit would _not_ be perpendicular to the velocity.

The answer to Joe's question is simple: The change in energy due to an
impulse is _second order_ in the size of the impulse, whereas the change in
momentum and change in direction are _first order_. So in the limit of a
large number of small impulses, all perpendicular to the velocity, you can
change the direction just fine without changing the energy.

Work it out: Start with an N-sided figure. There will be N triangles
consisting of (old momentum + impulse = new momentum). Don't assume the
impulse is perpendicular to the old momentum; assume the impulse is at
whatever angle is necessary to create a closed figure. Consider what
happens when we change this to a 2N-sided figure. The new triangles are
_not_ similar to the old triangles. They are twice as close to being right
triangles. Do some examples for yourself. Calculate the
angles. High-school-level geometry skills suffice. I recommend starting
with N=6; smaller N is confusing and larger N is just extra work.