What is thing called 'momentum'?
(Rhetorical question.)
The concept emerged in the first half the 17century.
The mechanical philosophers (e.g. Descartes, Huygens, Newton)
of this period gave us the practice of assigning to motion the quantity
known as 'momentum'. It prove to be a boon for the artillerist.
Before the momentum revolution, the concept of 'impetus'
dominated the science of motion, but it was of limited value
to the artillerist.
I think there is a meaningful parallel between the pre-classical
struggles to explain projectile motion and the state of modern
physics.
I also recommend reading the first chapter of the book _The Power of Images
in Early Modern Science_ (2003).
The essay,
The Challenging Images of Artillery:
Practical Knowledge at the Roots of the Scientific Revolution,
includes some nice prints and drawings and it sheds light on those
perplexing drawings of ballistic trajectories made before classical
mechanics.
Harry
leaking pen at [EMAIL PROTECTED] wrote:
> note though, that after velocity, they all become a measure of the
> rate of change. where is the original motion that is being constantly
> modified? there has to be an endpoint, yes no? or start point, as it
> were.
>
>
> On Tue, 22 Feb 2005 19:20:18 +0000, Grimer
> <[EMAIL PROTECTED]> wrote:
>> In previous posts the idea of a series of increasing
>> orders of derivatives of length with respect to time
>> has been chewed over.
>>
>> It will be helpful to recapitulate the discussion by
>> using excepts from the Vortex archives.
>>
>> On the subject of imagining what these derivative mean
>> physically, we have.
>>
>> ========================================================
>> ...Funny you should say that, Richard, because I've been
>> pondering how one could physically visualize high order
>> derivatives of distance with respect to time.
>>
>> dL/dT ......VELOCITY .......moving scenery
>> - no problem
>>
>> d2L/dT2 ....ACCELERATION ...being pushed back in ones
>> seat as the plane takes off
>> - no problem
>>
>> d3L/dt3 ....JERK............Mmm..more difficult - being
>> hit over the head with a
>> bottle perhaps?
>>
>> d4L/dT4 ....JOUNCE..........I have no feeling whatsoever
>> for this or high derivatives.
>>
>> But the failure to visualise these higher order derivative
>> is because I am thinking in terms of straight line motion.
>> If I think instead in terms of circular motion, or better
>> still, helical motions, then things become very much easier.
>>
>> If I allow myself to be pinned to the wall of a fairground
>> centrifuge then I can experience being "pushed back in my
>> seat on a continuous basis. By imposing a circular motion
>> on this circular motion to form an open vortex helix I can
>> visualize the next derivative, though I am well past the
>> age where I would want to experience it - and so on - and
>> so forth.
>>
>> ========================================================
>>
>> On the topic of visualisation Keith Nagel wrote,
>>
>> ======================================================
>> Also, you mentioned Jerk and Jounce ( sounds like
>> a b-list rap group ). I've also puzzled over the
>> physical meaning of these terms. It's rather like
>> trying to imagine higher dimensional shapes. One
>> dimension up is about all I can muster, which in
>> this case is Jerk. Standing on a carousel, with
>> the speed increasing and decreasing sinusoidally,
>> ought to do it. Perhaps a better term would be
>> "projectile vomiting" rather than jerk, huh??? (grin).
>> ======================================================
>>
>> The discussion then wandered off into considering the
>> outcome of experiments with coil-coils and coil-coil....
>> coils - and a host of other topics - as discussions on Vortex
>> frequently do <g> - which is, of course, one of the delights
>> of the Group. One is not artificially constrained to narrowly
>> keep to the point. 8-)
>>
>> I can now see another way of visualizing the higher order
>> differentials but before doing that I would like to state
>> something I have realised in relation to the conservation
>> laws.
>>
>> We have:-
>>
>> =======================================================
>> SYMBOL DERIVATIVE PROPERTY CONSERVED
>>
>> dL/dt velocity momentum yes
>>
>> d2L/dt3 acceleration energy yes
>>
>> d3L/dt3 jerk angular
>> acceleration yes
>>
>> d4L/dt4 jounce rate of change Err...?
>> of ang.acc.
>>
>> d5L/dt5 Err...? RoC of RoC Err...?
>> of ang.acc.
>>
>> ....... ..... ........... .......
>>
>> dnL/dtn ...
>>
>> =======================================================
>>
>> It has become increasingly clear to me that each derivative
>> is associated with a conservation law of its own. In short
>> there are an indefinitely large numbers of conservation
>> laws of motion. The reason we fail to see this is that we
>> are hag ridden by Cartesian Geometry and its unbounded x,
>> y and z dimensions. The coil-coil...coil visualization
>> discussed previously on Vortex avoided this trap because
>> it implicitly involved upper and lower bounds to the size
>> of the coils.
>>
>> It is interesting to note that the calculus also avoids
>> the Cartesian trap by not associating length with the
>> Cartesian space. The calculus allows us to have any
>> number of independent dimensions since no finite amount
>> of the (n+1)th derivative will give us a nth derivative.
>>
>> In the Cartesian case we can move from x (length) to
>> y (area) to z (volume)- but beyond that we are stuffed.
>>
>> In Keith's post he got as far as the third derivative
>> (angular acceleration) by "Standing on a carousel, with
>> the speed increasing and decreasing sinusoidally,..."
>> but could take it no further.
>>
>> I have now discovered how to generate physical
>> equivalents of d4L/dt4 and all higher derivative
>> motions with devices that anyone who is familiar
>> with the action of gyroscopes can readily visualise
>> and understand.
>>
>> Consider the Mechanism set out in
>>
>> http://tinyurl.com/4qc9g
>>
>> This is a gyro consisting of an outer ring of 8 gyros.
>> The eight outer ring gyros (2nd order gyros) are free
>> to rotate around their extended axles. These axles are
>> fixed rigidly to the central hub so that the rotating
>> 2nd order gyros are forcibly constrained to follow a
>> circular path when the central hub is rotated.
>>
>> Now I have chosen 8 gyros for pedagogic purposes since
>> it is easy to visualise the solid outer ring of a
>> classic gyro being replaced by multiple small gyros.
>> However, to take the physical example to d4L/dt4 and
>> higher order derivative motions it is easier to think
>> in terms of only a single pair of opposed gyros.
>>
>> A diagrammatic representation of a 7 order device
>> is shown in:
>>
>> http://tinyurl.com/6qus9
>>
>> As you can see, it makes a rather nice
>> Iterative Hierarchical (IH) fractal pattern.
>> The different orders are indicated by different
>> colours.
>>
>> At this point it is worth repeating something from
>> a previous post. Cartesian geometry as taught at
>> school is a mathematical abstraction. It can't be
>> applied willy-nilly to the real world. In that
>> world real things have upper and lower bounds so one
>> must use an x, y, and z geometry which also has upper
>> and lower bounds. This means that we can nest as
>> many spaces as we like in the manner of Russian dolls
>> or Chinese boxes.
>>
>> In this case the upper and lower bounds of IH orders
>> are quite obvious if you think about it. A gyro of
>> the nth order is bounded in size by the gyros of the
>> (n+1)th and the (n-1)th orders. Likewise with every
>> other order. It's precisely the same situation as
>> with the "coiled-coiled-coiled-........coil" mentioned
>> in a previous post. Like the different order of gyros,
>> each different order of coils implicitly has an upper
>> and lower bound since a coil cannot be smaller than
>> twice the diameter of the wire, nor larger than the
>> next coil up the pecking order.
>>
>> Interestingly enough the IH fractal can be seen as
>> an attenuated version of the turbulent structure of
>> water. In the words of the poet,
>>
>> ====================================
>> Big whorls have little whorls
>> That feed on their velocity,
>> And little whorls have lesser whorls
>> And so on to viscosity.
>>
>> - Lewis F. Richardson -
>> ====================================
>>
>> And the power laws discovered for water vapour
>> are no doubt a reflection of their fractal
>> dynamic structure.
>>
>> Cheers
>>
>> Frank Grimer
>>
>>
>
