Re: Seeming paradox


"Eyal Lotem" <eyal.lotem@xxxxxxxxx> wrote in message news:ev7s90$mn9$1@xxxxxxxxxxxxxxxxxxxxxxxxx
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


There's your problem. We can NOT say "twice".
mv^2 = 2 * 1/2 mv^2 =/= 1/2m(2v)^2

Then the problem is compounded by the loss of mass of the heavy ball.

Energy is relative because v is relative. The heavy ball fired
by the third ship goes backwards relative to the second ship,
and may go backwards, stand still or forwards relative to the first ship.

Let's put it this way.
An empty truck of mass 1 (tonne or ton if you like, it doesn't matter)
has a road velocity of 100 mph (or kilometers/hour, the units don't
matter) and crashes into a tree with just enough energy to knock
the tree over, coming to a complete stop.

Energy needed is 1/2 * 1 * 100^2 = 5000.
How fast must the same truck now carrying a load of 1 ton be
going to knock over the same tree?
Is it:
a) 100 mph?
b) 50 mph?
c) 10 mph = sqrt(100) mph?
d) other?

Energy needed to fell the tree is 1/2 * 2 * v^2 = 5000.
Answer below
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71 mph.

.

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