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details of the gas motions in the cap explosion, yet we could predict the velocities
with which the bodies came apart, for example. Another interesting example is
rocket propulsion. A rocket of large mass, ă, ejects a small piece, oŸ mass mm,
with a terrific velocity V relative to the rocket. After this the rocket, If it were
originally standing still, will be moving with a smaill velocity, ø. sing the
principle of conservation of momentum, we can calculate this velocity to be
b=T: V.
So long as material is being ejected, the rocket continues to pick up speed. Roecket
propulsion is essentially the same as the recoil of a gun: there is no need for any
air to push against.
10-5 Relativistic momentum
In modern times the law oŸ conservation of momentum has undergone certain
modifcations. However, the law is still true today, the modifications being mainly
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in the defñnitions of things. In the theory of relativity it turns out that we do have
conservation of momentum; the particles have mass and the momentum ïs still
given by ?m0, the mass times the velocity, bu the rmass changes tuïth the uelocit,
hence the momentum also changes. The mass varies with velocity according to
the law
m= TT —0——. (10.7)
V1=u2/°
where ?nọ is the mass of the body at rest and e is the speed of light. It is easy to
see from the formula that there is negligible diference between mm and mo unless
ò is very large, and that for ordinary velocities the expression for momentum
reduces to the old formula.
The components of momentum for a single particle are written as
Tĩì0U„ ThoUụ TH0uUz
mm... ——-. ¬=a.. .
where øŸ = 02+ D + 02. IÝ the z-components are summed over all the interacting
particles, both before and after a collision, the sums are equal; that is, momentum
1s conserved in the z-direction. The same holds true in any direction.
In Chapter 4 we saw that the law of conservation of energy is not valid unless
we recognize that energy appears in diferent forms, electrical energy, mechanical
energy, radiant energy, heat energy, and so on. In some of these cases, heat
energy for example, the energy might be said to be “hidden.” 'This example
might suggest the question, “Are there also hidden forms of momentum——perhaps
heat momentum?” 'Phe answer is that it is very hard to hide momentum for the
following reasons.
The random motions of the atoms of a body furnish a measure of heat energy,
1f the sguares of the velocities are summed. 'Phis sum will be a positive result,
having no directional character. The heat is there, whether or not the body moves
as a whole, and conservation oŸ energy in the form of heat is not very obvious.
On the other hand, 1ƒ one sums the 0eloczfzes, which have direction, and fñnds a
result that is not zero, that means that there is a drift of the entire body in some
particular direction, and such a gross momentum is readily observed. Thus there
is no random internal lost momentum, because the body has net momentum only
when i% moves as a whole. 'Therefore momentum, as a mechanical quantity, 1s
difcult to hide. Nevertheless, momentum cøø be hidden-—in the electromagnetic
ñeld, for example. 'This case is another efect of relativity.
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One of the propositions of Newton was that interactions at a distance are
instantaneous. Ït turns out that such is not the case; in situations involving elec-
trical forces, for instance, 1ƒ an electrical charge at one location is suddenly moved,
the efects on another charge, at another place, do not appear instantaneousÌy——
there is a little delay. In those circumstances, even If the forces are equal the
momentum will not check out; there will be a short time during which there will
be trouble, because for a while the first charge will feel a certain reaction force,
say, and will picek up some momentum, but the second charge has felt nothing
and has not yet changed its momentum. lt takes time for the inÑuence tO cross
the intervening distance, which it does at 186,000 miles a second. In that tiny
time the momentum of the particles is not conserved. OÝ course after the second
charge has felt the efect of the first one and all is quieted down, the momentum
cequation will check out all right, but during that small interval momentum is not
conserved. We represent this by saying that during this interval there is another
kind of momentum besides that of the particle, mu, and that is momentum in
the electromagnetic field. If we add the feld momentum to the momentum of
the particles, then momentum is conserved at any moment all the time. “The
fact that the electromagnetic field can possess momentum and energy makes that
fñeld very real, and so, for better understanding, the original idea that there are
Jjust the forces bebween particles has to be modified to the idea that a particle
makes a field, and a field acts on another particle, and the field itself has such
familiar properties as energy content and momentum, just as particles can have.
To take another example: an electromagnetic fñeld has waves, which we call light;
it turns out that light also carries momentum with it, so when light impinges
on an object 1% carries in a certain amount of momentum per second; this is
equivalent to a force, because if the illuminated object is picking up a certain
amount of momentum per second, its momentum is changing and the situation
1s exactly the same as If there were a force on it. Light can exert pressure by
bombarding an object; this pressure is very small, but with sufficiently delicate
apparatus it is measurable.
Now in quantum mechanics it turns out that momentum is a diferent thing——
1E is no longer rm0. It is hard to defñne exactly what is meant by the velocity oŸ a
particle, but momentum still exists. In quantum mechanies the diference is that
when the particles are represented as particles, the momentum 1s still ru, but
when the particles are represented as waves, the momentum is measured by the
number of waves per centimeter: the greater this number of waves, the greater
the momentum. In spite of the diferences, the law of conservation of momentum
--- Trang 211 ---
holds also in quantum mechanics. Even though the law #! = rma is false, and
all the derivations of NÑewton were wrong for the conservation oŸ momentum, in
quantum mechanics, nevertheless, in the end, that particular law maintains itselfl
--- Trang 212 ---
Weoe£or-s
11-1 Symmetry in physỉcs
In this chapter we introduce a subject that is technically known in physics
as sumưmnetrụ tín phụsical lau. The word “symmetry” is used here with a special
meaning, and therefore needs to be defñned. When is a thing symmetrical—how
can we defne it? When we have a picture that is symmetrical, one side 1s
somehow the same as the other side. Professor Hermann Weyl has given this
defnition of symmetry: a thing is symmetrical iŸ one can subject it to a certain