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### 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/
```
Associated Topics:
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