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object, as shown in Eig. 16-4(a). The mass ?n of each corresponds to +0, which, as
we know, is ?mo/4/1 — w2/c2. T we assume the conservation oŸ momentum and
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"5.1. ——:
(@) — M mạ *Ủ @
Fig. 16-4. Two views of an Inelastic collision between equally massive
objects.
the principle of relativity, we can demonstrate an interesting fact about the mass
of the new object which has been formed. We imagine an infinitesimal velocity u
at right angles to œ0 (we can do the same with ñnite values of œ, but iE is easier to
understand with an infinitesimal velocity), then look at this same collision as we
ride by in an elevator at the velocity —w. What we see is shown in Fig. 16-4(b).
The composite object has an unknown mass M. Now object 1 moves with an
upward component of velocity œ and a horizontal component which is practically
cqual to +œø, and so also does object 2. After the collision we have the mass Ä⁄ƒ
moving upward with velocity œ, considered very small compared with the speed
of light, and also small compared with +. Momentum must be conserved, so let
us estimate the momentum ¡in the upward direction before and after the collision.
Before the collision we have p 2m„„u, and after the collision the momentum is
evidently ø' = M„u, but Ä⁄„ is essentially the same as Äíức because œ is so small.
These momenta must be equal because of the conservation of momentum, and
therefore
Mẹ = 2m. (16.11)
The rnass oƒ the object thích ¡s ƒormecd hen tuo equal objects collide rmmust be
tuñce the mmass 0ƒ the objects tuhích come together. You might say, “Yes, oŸ course,
that is the conservation of mass.” But not “Yes, of course,” so easily, because
these mmasses hœue been enhanced over the masses that they would be if they were
standing still, yet they still contribute, to the total Ä, not the mass they have
when standing still, but more. Astonishing as that may seem, in order for the
conservation of momentum to work when two objects come together, the mass
that they form must be greater than the rest masses of the objects, even though
the objects are at rest after the collision!
16-5 Relativistic energy
In the last chapter we demonstrated that as a result of the dependence of
the mass on velocity and Newton's laws, the changes in the kinetic energy oŸ an
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object resulting from the total work done by the forces on it always comes out to
AT' = (mụ — mọ)c? = ——9_- mạọc?. (16.12)
V1— u2/e°
We even went further, and guessed that the total energy is the total mass tỉmes cẺ.
Now we continue this discussion.
Suppose that our bwo equally massive objects that collide can still be “seen”
inside Mĩ. Eor instance, a proton and a neutron are “stuck together,” but are still
moving about inside of Mƒ. Then, although we might at fñrst expect the mass MỸ to
be 2m, we have found that it is not 2mọ, but 2?n¿„. Since 2?n„„ is what is put ïn,
but 2mọ are the rest masses of the things inside, the ezcess mass of the composite
obJject is equal to the kinetic energy broughtin. This means, of course, that energu
has zmertia. In the last chapter we discussed the heating of a gas, and showed that
because the gas molecules are moving and moving things are heavier, when we put
energy into the gas its molecules move faster and so the gas gets heavier. But in
fact the argument is completely general, and our discussion of the inelastie collision
shows that the mass is there whether or not i is knetc energy. In other words, if
two particles come together and produce potential or any other form of energy; if
the pieces are slowed down by climbing hills, doing work against internal Íorces, or
whatever; then it is still true that the mass is the total energy that has been put in.
So we see that the conservation of mass which we have deduced above is equivalent
to the conservation of energy, and therefore there is no place in the theory of
relativity for strictly inelastic collisions, as there was in Newtonian mechanics.
According to Newtonian mechaniœs it is all right for two things to collide and so
form an object of mass 2mọ which is in no way distinct from the one that would
result from putting them together slowly. Of course we know from the law of
conservation of energy that there is more kinetic energy ¡nside, but that does not
affect the mass, according to Newton”s laws. But now we see that this is impossible;
because of the kinetic energy involved in the collision, the resulting object will
be heauier; therefore, it will be a đjƒerent object. When we put the objects
together gently they make something whose mass is 2m; when we put them
together forcefully, they make something whose mass is greater. When the mass is
diferent, we can #ell that it is diferent. 5o, necessarily, the conservation of energy
must go along with the conservation of momentum in the theory of relativity.
This has interesting consequences. For example, suppose that we have an
object whose mass Ä⁄ƒ is measured, and suppose something happens so that it fies
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into two equal pieces moving with speed +, so that they each have a mass Tn¿„.
Now suppose that these pieces encounter enough material to slow them up until
they stop; then they will have mass mọ. How much energy will they have given
to the material when they have stopped? Each will give an amount (m„„ — mọ)€Ẻ,
by the theorem that we proved before. 'Phis much energy is left in the material
in some form, as heat, potential energy, or whatever. Now 2m¿„ —= M, so the
liberated energy is # = (ÁMf — 2mo)c?. This equation was used to estimate how
much energy would be liberated under fssion in the atomic bomb, for example.
(Although the ragments are not exactly equal, they are nearly equal.) The mass
of the uranium atom was known——it had been measured ahead of time—and the
atoms into which ït split, iodine, xenon, and so on, all were of known mass. By
masses, we do not mean the masses while the atoms are moving, we mean the
mmasses when the atoms are ø res. In other words, both MỸ and rmọ are known.
So by subtracting the two numbers one can calculate how much energy will be
released 1f Mƒ can be made to split in “half” Eor this reason poor old Einstein
was called the “father” of the atomie bomb in all the newspapers. Of course, all
that meant was that he could tell us ahead of time how much energy would be
released if we told him what process would occur. The energy that should be
liberated when an atom of uranium undergoes fission was estimated about six
months before the frst direct test, and as soon as the energy was in fact liberated,
Someone measured it directly (and if Einsteins formula had not worked, they
would have measured it anyway), and the moment they measured it they no
longer needed the formula. Of course, we should not belittle Einstein, but rather
should criticize the newspapers and many popular descriptions of what causes
what in the history of physics and technology. The problem of how to get the
thing to occur in an efective and rapid manner is a completely diÑferent matter.
The result is just as significant in chemistry. Eor instance, if we were to weigh
the carbon dioxide molecule and compare its mass with that of the carbon and