E1 The Symmetric Punch
The English and the French were for many years on the warpath with one another. They not only fought politically and militarily, but they also argued about whether the ‘punch’ (lat. ‘impetus’) of a projectile increased linearly with speed or as the square of the speed. On the island a vector-oriented viewpoint was preferred, and accordingly ‘momentum’ was described by p = m • v. On the continent one rather relied on scalar values such as
E = 0.5 • m • v2. These preferences still show up today in colloquial language: If an Englishman must determine a value ‘then he will figure it out’ (tending to draw a geometric figure), while the Frenchman ‘va calculer ça’ (tending to solve some algenbraic equations).
Today we know that both, momentum and kinetic energy have meaning, and therefore it is not surprising that both the French and the English got correct results. This example shows very nicely, how the physical terminology needed to gradually crystallize and did not already possess their current definitiveness when they were first introduced. It is precisely these terms, i.e. (inertial) mass, momentum and kinetic energy, which are the subject of this chapter.
We must first speak of momentum, which we maintain to be defined by p = m • v ! As soon as it is clarified how inertial mass is transformed from one reference system to another, then it is also clear what happens to the momentum, because we already know the transformation of the velocities in D4 and D5. In any case it should be clear that momentum, just like velocity, is a highly relative quantity whose value (not only in the STR !) depends entirely on the choice of reference system: In the system of a body at rest its momentum is always zero.
Our derivation follows the beautiful presentation in [15-110ff]. This presentation is beautiful because it deduces the equality of two values based on the symmetry of a thought experiment. The twins Peter and Danny are each standing on one of two Einstein trains, which are racing past each other in opposite directions on a long straight stretch of track. They each throw completely symmetrical punches at each other perpendicular to the direction of travel of the trains:
The symmetry of the situation does not allow one twin to punch more strongly than the other. Both fists (measured at rest!) are of equal weight and both fists are equally fast in their own system (u' = -w). At any time the sum of the impulses in the y-direction, that is, perpendicular to the direction of travel, is zero for both. We regard the impact with Peter in the black, non-prime system and see the perpendicular velocity of u' of Danny's fist slowed according to D5. That Danny's fist carries nevertheless an equally large momentum as Peter’s can only be explained in that Danny has somehow increased the mass of his fist. Here Epstein tells the story of the aging bouncer in Bourbon Street, who can no longer punch as fast as former times and compensates by putting a role of coins into his fist, in order to give his punch its former strength…
We soberly calculate assuming that the mass depends on the relative velocity. We use v (as always) to show the relative velocity in the x-direction, that is, the direction of travel of the trains:
|py (Danny) = - py (Peter)||due to symmetry for Peter and for Danny|
|mv+u • u = - mw • w||for Peter and us! u = u' • √ = - w • √ (according to D5) yields|
|mv+u • (- w • √ ) = - mw • w xxxxxxxxxxx||dividing through by - w we obtain|
|mv+u • √ = mw||where v and u are perpendicular and are added as vectors!|
This relationship applies for arbitrarily small perpendicular speeds of u' = -w, and hence for the limiting case of u' = -w = 0. In that case we have u = 0 , mv+u = mv and mw = m0 . The term mw = m0 represents the mass of the fist at rest, the so-called rest mass. Thus we have found the dependence of the mass on relative velocity v: The inertial mass of a body increases with relative velocity, giving
This ‘dynamic mass’ mv has to be used for the relativistic momentum:
Thus the mass of a particle increases with its velocity. Without allowing for this relativistic effect modern particle accelerators could not function! The mass increase is dramatic when the particle approaches the speed of light: The relationship mv/m0 follows the function 1/√, and as v → c this function approaches infinity:
We now have a completely new argument for c as a limiting velocity: The mass being accelerated increases infinitely, as the velocity v approaches the speed of light! The body thus offers an ever larger resistance to further acceleration. We will see this even better when we treat the question of energy.