Figure 4.2: whatever comes in, the calorimeter generates only standard impulse carriers Who would think that at the billiard table one acts as a physicist? When Bob strikes he is dealing with interactions of motion. He begins to behave as a physicist if he steers and sorts these interactions in a controlled way. We illustrate the construction of an experi-mental instrument for a basic measurement of ”capability to work” (energy) and ”impact”
(momentum){4.3}. We construct a Gedanken-model on the operation of a particle detector:
a calorimeter (see figure 4.2 with internal process as black box). It contains a reservoir with identically constituted elements{1 v=0}which are initially at rest. An incoming particlea v with velocityv will be slowed downa v=0 by successive collisions with the resting elements from the reservoir (see figure 4.1b). While in a generic billiard collision all object balls fly off with arbitrary impetus, now the process is set up and controlled from outside. A team of assistants will steer the process. As a result of stopping the incident cue ball a certain number of reservoir elements {1 v1} (with standardized velocity v1) will be knocked out of the calorimeter. Each ball has exactly the same impact behavior and represents a unit of momentum. In this model we can count the number of extracted reference units. The ele-mental ordering relations ”more capability to work than” (energy) and ”more impact than”
(momentum) become measurable.
We introduce the following notation for tangible objects
• a−→v for a moving particle with individual namea
• a v=0 or simply a 0 when the object rests and in formal mathematical expressions:
• a x,v for a particle a at the place xwith velocity v
• a v when the place does not matter, similarly for a moving standard springS
v etc.
• a v=0 ora 0 and S
0 when the respective object rests.
4.2 Basic observable
From daily work experience and in play (where valid prognoses about natural processes pay off most) one cannotice the ”impact” of decelerating bodies and the associated ”capability to
work”. One developspre-theoretic comparison methods by examples and in words. Physicists fix theconventions for observer independent and reproducible procedures. Without viewing motion as a mere mathematical map γ :τ →(t,x) ; we specify comparison methods so that
• the procedure is universally available and
• the result does not depend on the individual observer; all physicists find the same [34].
For collisions of irrelevant inner structure Galilei defines an elementaryordering criterion.
Let two generic bodiesa andb run into each other with initial velocitiesva resp. vb, collide and stick together.
Definition 3 Momentum p[a v] is the striking power, impact (Wucht) of the moving body a v [12]. Object a va has more impact than object b vb
a va >P b vb (4.1)
if in a head-on collision test one body overruns the other.
If the bound aggregate a va,b vb ⇒ a ∗ b v=0 moves neither with object a va to the right nor with object b vb to the left, then their impact is practically the same p[a va] =p[b vb].
Definition 4 Inertia m[a] is the - passive - resistance against changes of the state of motion of object a [3] [23]. According to Galilei object a is more massive than object b
a >m b (4.2)
if after an inelastic head-on collision testa v,b−v ⇒ a ∗ b v with same initial velocity the bound composite moves in the direction v ∝v of the heavier object [18].
Remark 4 One conducts a special case of impulse comparison
>m := >P
va=−vb
with an extra condition that both objects initially move head-on with same velocity va =! −vb. Consider a bow B and a crossbow C. When the string is tightened (and mechanically locked) the charged devices BE,CE become energy sources. With the charged state we asso-ciate a capability to work (Wirkungsverm¨ogen). According to Leibniz equipollence principle (detailed discussion in {4.6.1}) we compare it indirectly by measuring their (kinetic) effect against the same test system; whether the same test particles (e.g. archerG1 and arrowG2 in figure 4.3) repulse with larger velocity Δv1,2 >Δv1,2 than from a shot with the weaker bow.
Without restricting generality the charged sources (crossbow CE
0, bow BE
0) may initially be at rest and after pulling the trigger, after expending the associated capability to work both discharged sources C, B remain at rest. This allows a clear separation. In return the
4.2. Basic observable 47
Figure 4.3: a) charged bowBE or crossbowCE represent energy sources which can be coupled into b) a two-body system of initially resting archer and arrow G1 ∪ G2 c) charged bow BE
expends its energy on test system G1∪ G2 (discharged bow B
0 remains at rest) d) charged crossbow CE causes a larger kinetic effect Δvi >Δvi against the same test system G1 ∪ G2
test particles (archer G1 and arrow G2) begin to fly apart. With their motion we associate another form of capability to work (kinetic energy), which they can expend against third parties etc. According to Helmholtz measurement principle: the total capability to work is conserved. In a measurement we consume one specific form entirely (e.g. potential energy until a spring is entirely relaxed; kinetic energy until all projectiles come to rest etc.) in a transformation into other forms (preferably carried by separate elements of the system).
Definition 5 Energy is the capability of a separate source or system to work against an external system G. The kinetic, potential, binding etc. form of energy is associated with exhausting a particular condition of the source (motion, configuration size, chemical bound).
According to Leibniz one source
SE >E S˜E˜ (4.3)
has more potential than another source S˜E˜ if the effect of source SE on the same test system G exceeds the effect of source S˜E˜.
In our calorimeter model we will measure the kinetic energy of a moving body a va by the number of obstacles it overcomes. We count how many standard springs can be compressed (repeat elementary processes) before the body a 0 stops. The kinetic energy (of projectiles) transforms into potential energy (of the absorber material). If the latter comes in standard portions, which are all congruent with one another, our quantification is complete.
We define the practical comparison circularity free, without presupposing numerical val-ues (on a ratio scale) of unclear origin and status. Physics goes beyond the formal equations:
Mathematics postulates its abstracta as known and given; instead we scrutinize the formation from tangible operations. Under the abstraction ”energy” and ”momentum” one compares two interactions with regards to an elemental ordering criterion.
Definition 6 In an abstraction we regard the common quality of both objects for itself with-out needing to consider the dissimilarity (of both objects in other regards).2
The comparison procedure distinguishes one singular aspect (kinetic effect) for observation.
We regard all objects (moving billiard ball, compressed spring, battery etc.) solely as sub-stitutable carriers of their common quality ”capability to work” and ”impact”.3
2Helmholtz [7] explains ”bodies whose weight we are comparing can be made from most different materials, different shape and volume. The weight - which we set equal - is only one of their properties and obtained by abstraction. We are only justified - to call those bodies themselves weights and designate these weights as quantities - in circumstances where we can disregard all other properties of these bodies”.
Ruben [22] thinks about: ”a tree e.g. is in general a subject of biology. If the tree is cut down then it is - for the worker who has to get out of the way of the falling tree - a mechanical object. In this context the tree is essentially important as carrier of weight; it is unimportant whether the tree is a linden or an oak.
All natural things are always also carrier of mass. Insofar as they are they are a subject of mechanics.”
3In the theory one makes propositions about abstract ”energy” and ”momentum”. The transition is implemented by a limitation in the manner of speaking onto invariant assertions [21]. One restricts from simple descriptive sentences (about various colloquial aspects of an interaction) onto assertions, which remain unchangedly valid under substituting equivalent energy-momentum carriers; e.g. an equally charged crossbow CE and bowBE rebound all test particles in the same way (despite different inner structure or materials).
4.3. Quantification 49