Conservation of Kinetic Energy

Date: 6/8/96 at 3:56:22
From: PB
Subject: The conservation of kinetic energy under a translational 
transformation

Dear Sir,

I am a mechanical engineer from Sweden, and I got stuck with a 
theoretical questions in physics.  

The formula mv^2/2 + mgh = constant is the energy conservation formula 
in physics, and is said to be invariant for all observers who move 
with constant velocity relative to one another. Suppose that I am an 
observer who moves with the body: then v = 0. So how is this formula 
conserved?

John Mendelsson

Date: 6/8/96 at 7:5:23
From: Doctor Anthony
Subject: Re: The conservation of kinetic energy under a translational 
transformation

The energy conservation formula refers to a frame of reference.  If 
the frame of reference is the body itself, then v=0, h=0 and this 
simply makes the constant equal to 0.  Total energy is still 
conserved.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 6/8/96 at 9:55:26
From: PB
Subject: Re: The conservation of kinetic energy under a translational 
transformation

I think the formulation of my question  does not reflect my thoughts. 
I give another example:  the work energy theorem: I Fds = mv^2/2  
where "I" is the integral sign (I cannot produce it with my software). 
Suppose that another observer, B, who is moving 5 m/s faster than I 
am, watches the work done and measures m(v+5)^2/2-mv^2/2, which is not 
equal to mv^2/2, whereas I Fds is still the same.... or isn't it ?

Date: 6/13/96 at 21:36:48
From: Doctor Tom
Subject: Re: The conservation of kinetic energy under a translational 
transformation

Energy is NOT conserved under a change of reference frame; your
example is a perfect demonstration of that.

When you change reference frames, some things are conserved, and
some things are not.  One of the most interesting things in physics 
studies is which things are conserved, and which are not.

For example, in Newtonian physics, at a fixed time, two observers
of two particles will observe the same distance between them, but in 
Einstein's relativistic physics, they will not.  However, in 
Einstein's physics, they will observe the same "interval" between 
events, where the "interval" includes a term involving the observed 
time of the events.

Two Newtonian observers would notice that light travels at different 
speeds in different reference frames.  Einsteinian observers will see 
no difference.  And the real world appears as the Einsteinian 
observers would predict.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 6/22/96 at 10:2:9
From: peter
Subject: Re: The Conservation of Kinetic Energy under a Translational 
Transformation

In a previous response, you explained that the ds (velocity) used in 
determining work done depends on the reference frame; my mistake was 
that I thought that ds is constant, which it is not in this case, 
since here the particle accelerates due to a force F, i.e. has not 
const velocity. But since you are a mathematician, you ought to be 
able to prove it purely  mathematically, not just make it plausible.
Actually I feel that all Newtonian mechanics could be made pure math, 
i.e. formulated as axioms, although I do not know how this is done, 
nor have I seen any book introducing these ideas...

Date: 6/24/96 at 14:13:34
From: Doctor Tom
Subject: Re: The Conservation of Kinetic Energy under a Translational 
Transformation

My example can be made mathematically rigorous - I just find it much
easier to understand using physical intuition.  You can certainly
prove that the laws of physics will appear the same in different
Galilean frames from first principles.

I've never found it useful to work physics problems from "axioms",
although I suppose it might be possible to write down those axioms.

I have always had a wonderful "mathematical intuition", but my 
"physical intuition" has never been as good, and I've had to struggle 
to get what little I have.  I know lots of physicists who are exactly 
the opposite.  I suspect the world's best physicists (and 
mathematicians) are those with strong intuitions in both areas.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/