Seeming paradox
- From: Eyal Lotem <eyal.lotem@xxxxxxxxx>
- Date: Sat, 07 Apr 2007 13:37:19 +0300
Hi.
My explanation is a little long, but I think it is simple, and I tried
making it as clear as I can. It should only take a few minutes to explain,
and I am greatly puzzled by this. Thank you!
I was wondering about an apparent discrepancy in simple low-speed Newtonian
physics.
The source of the issue is that Kinetic Energy, being a function of the
squared velocity, is relative to the viewer's speed (because the velocity
is relative to the viewer's speed), while other forms of energy (for
example potential energy in a spring) are not relative to the viewer's
speed.
A mind experiment can be used to explain the apparent paradox, and I'd be
grateful if someone explained the underlying mistake in their assumptions:
The 3 ships.
3 ships of the same weight are flying at the same velocity in space, each
carrying 2 heavy ball weights attached to a spring.
The springs are a form of "engine" that can shoot out the heavy balls in
order to push forward the ship.
Initially, the 3 ships are all in the same state.
Then, two of the ships fire out one of the heavy balls (releasing energy E),
resulting in their acceleration to the speed of V, relatively to the ship
that did not fire.
Then, one of the two ships fires out another heavy ball (releasing energy
E), resulting in its acceleration to the speed of V', relatively to the
ship that did not fire.
Now, looking at the experiment as a whole, we can say that the ship who
fired 2 balls gained twice the kinetic energy of the ship who fired 1 ball
(and is a little lighter now that it does not contain both balls) - and
therefore should travel a little faster than sqrt(2)*V (i.e V' >=
sqrt(2)*V).
However, if one examines the two events *separately*, the conclusions seem
to be altogether different.
If the experiment of the 2 ships is done separately from the one of the 3
ships, or at least examined separately, then we can "forget" that the 2
ships already accelerated to the speed of V. Then it would seem that if one
of the two ships is firing E energy, it should accelerate by more than V
(since its lighter) compared to the ship that isn't shooting its available
ball.
So a viewer who viewed the whole experiment would think that the ship speeds
are 0, V, and a bit more than sqrt(2)*V.
A viewer who viewed just the part with two ships, one of which shoots
another ball, would think that the two ship speeds are 0, V.
That is a paradox, because the relative speed between the last 2 ships is a
bit more than V*(sqrt(2)-1) by one viewer, and a whole V*1 by the other
viewer!
The same spring energy (according to any viewer) converts to kinetic energy
that has a different meaning in velocity, depending on the viewer!
How is this possible?
Thanks,
Eyal
.
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