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Unformatted text preview: 10 Conservation of Momentum 10—1 Newton’s Third Law On the basis of Newton’s second law of motion, which gives the relation
between the acceleration of any body and the force acting on it, any problem in
mechanics can be solved in pr1nc1ple. For example, to determine the motion of a
few particles, one can use the numerical method developed in the preceding chapter.
But there are good reasons to make a further study of Newton’s laws. First, there
are quite simple cases of motion which can be analyzed not only by numerical
methods, but also by direct mathematical analysis. For example, although we
know that the acceleration of a falling body is 32 ft/secz, and from this fact could
calculate the motion by numerical methods, it is much easier and more satisfactory
to analyze the motion and ﬁnd the general solution, s = so + vol + l6t2. In
the same way, although we can work out the positions of a harmonic oscillator by
numerical methods, it is also p0551ble to show analytically that the general solution
is a simple cosine function of t, and so it is unnecessary to go to all that arithmetical
trouble when there is a simple and more accurate way to get the result. In the
same manner, although the motion of one body around the sun, determined by
gravitation, can be calculated point by point by the numerical methods of Chapter
9, which show the general shape of the orbit, it is nice also to get the exact shape,
which analysis reveals as a perfect ellipse. Unfortunately, there are really very few problems which can be solved exactly
by analysis. In the case of the harmonic oscillator, for example, if the spring force
is not proportional to the displacement, but is something more complicated, one
must fall back on the numerical method. Or if there are two bodies going around
the sun, so that the total number of bodies is three, then analysis cannot produce a
simple formula for the motion, and in practice the problem must be done numeri
cally. That is the famous threebody problem, which so long challenged human
powers of analysis; it is very interesting how long it took people to appreciate
the fact that perhaps the powers of mathematical analysis were limited and it
might be necessary to use the numerical methods. Today an enormous number of
problems that cannot be done analytically are solved by numerical methods, and
the old threebody problem, which was supposed to be so difﬁcult, is solved as a
matter of routine in exactly the same manner that was described in the preceding
chapter, namely, by doing enough arithmetic. However, there are also situations
where both methods fail: the simple problems we can do by analysis, and the
moderately difﬁcult problems by numerical, arithmetical methods, but the very
complicated problems we cannot do by either method. A complicated problem is,
for example, the collision of two automobiles, or even the motion of the molecules
of a gas. There are countless particles in a cubic millimeter of gas, and it would
be ridiculous to try to make calculations with so many variables (about 1017—
a hundred million billion). Anything like the motion of the molecules or atoms of
a gas or a block or iron, or the motion of the stars in a globular cluster, instead of
just two or three planets going around the sun—such problems we cannot do
directly, so we have to seek other means. In the situations in which we cannot follow details, we need to know some
general properties, that is, general theorems or principles which are consequences
of Newton’s laws. One of these is the principle of conservation of energy, which
was discussed in Chapter 4. Another is the principle of conservation of momentum,
the subject of this chapter. Another reason for studying mechanics further is that
there are certain patterns of motlon that are repeated in many different circum 10—1 10—1 Newton’s Third Law 10—2 Conservation of momentum
10—3 Momentum is conserved!
10—4 Momentum and energy 10—5 Relativistic momentum stances, so it is good to study these patterns in one particular circumstance. For
example, we shall study collisions; different kinds of collisions have much in
common. In the ﬂow of ﬂuids, it does not make much difference what the ﬂUId is,
the laws of the ﬂow are similar. Other problems that we shall study are vibrations
and oscillations and, in particular, the peculiar phenomena of mechanical waves—
sound, vibrations of rods, and so on. In our discussion of Newton’s laws it was explained that these laws are a kind
of program that says “Pay attention to the forces,” and that Newton told us only
two things about the nature of forces. In the case of gravitation, he gave us the
complete law of the force. In the case of the very complicated forces between
atoms, he was not aware of the right laws for the forces; however, he discovered
one rule, one general property of forces, which is expressed in his Third Law, and
that is the total knowledge that Newton had about the nature of forces—the law
of gravitation and this principle, but no other details. This principle is that action equals reaction. What is meant is something of this kind: Suppose we have two small bodies,
say particles, and suppose that the ﬁrst one exerts a force on the second one,
pushing it with a certain force. Then, simultaneously, according to Newton’s
Third Law, the second particle will push on the ﬁrst with an equal force, in the
opposite direction; furthermore, these forces effectively act in the same line.
This is the hypothesis, or law, that Newton proposed, and it seems to be quite
accurate, though not exact (we shall discuss the errors later). For the moment
we shall take it to be true that action equals reaction. Of course, if there is a third
particle, not on the same line as the other two, the law does not mean that the total
force on the ﬁrst one is equal to the total force on the second, since the third particle,
for instance, exerts its own push on each of the other two. The result is that the
total effect on the ﬁrst two is in some other direction, and the forces on the ﬁrst
two particles are, in general, neither equal nor opposite. However, the forces on
each particle can be resolved into parts, there being one contribution or part due
to each other interacting particle. Then each pair of particles has corresponding
components of mutual interaction that are equal in magnitude and opposite in
direction. 10—2 Conservation of momentum Now what are the interesting consequences of the above relationship? Sup
pose, for simplicity, that we have just two interacting particles, possibly of different
mass, and numbered 1 and 2. The forces between them are equal and opposite;
what are the consequences? According to Newton’s Second Law, force is the time
rate of change of the momentum, so we conclude that the rate of change of momen
tum p 1 of particle l is equal to minus the rate of change of momentum p2 of particle
2, or dpl/dt = —dp2/dt. (10.1) Now if the rate of change is always equal and opposite, it follows that the total
change in the momentum of particle l is equal and opposite to the total change in
the momentum of particle 2; this means that if we add the momentum of particle
1 to the momentum of particle 2, the rate of change of the sum of these, due to
the mutual forces (called internal forces) between particles, is zero; that is d(p1 + p2)/dt = 0. (10.2) There is assumed to be no other force in the problem. If the rate of change of this
sum is always zero, that is just another way of saying that the quantity (171 + p 2)
does not change. (This quantity is also written mlvl + mgvz, and is called the
total momentum of the two particles.) We have now obtained the result that the
total momentum of the two particles does not change because of any mutual
interactions between them. This statement expresses the law of conservation of 102 momentum in that particular example. We conclude that if there is any kind of
force, no matter how complicated, between two particles, and we measure or
calculate mlvl + m2222, that is, the sum of the two momenta, both before and
after the forces act, the results should be equal, i.e., the total momentum is a
constant. If we extend the argument to three or more interacting particles in more com
plicated circumstances, it is evident that so far as internal forces are concerned, the
total momentum of all the particles stays constant, since an increase in momentum
of one, due to another, is exactly compensated by the decrease of the second,
due to the ﬁrst. That is, all the internal forces will balance out, and therefore
cannot change the total momentum of the particles. Then if there are no forces
from the outside (external forces), there are no forces that can change the total
momentum; hence the total momentum is a constant. It is worth describing what happens if there are forces that do not come from
the mutual actions of the particles in question: suppose we isolate the interacting
particles. If there are only mutual forces, then, as before, the total momentum of
the particles does not change, no matter how complicated the forces. On the other
hand, suppose there are also forces coming from the particles outside the isolated
group. Any force exerted by outside bodies on inside bodies, we call an external
force. We shall later demonstrate that the sum of all external forces equals the rate
of change of the total momentum of all the particles inside, a very useful theorem. The conservat1on of the total momentum of a number of interacting particles
can be expressed as mlvl + m2v2 + m3v3 + = aconstant, (10.3) if there are no net external forces. Here the masses and corresponding velocities
of the particles are numbered 1, 2, 3, 4, . . . The general statement of Newton’s
Second Law for each particle, f = 55¢va (10.4) is true speciﬁcally for the components of force and momentum in any given direc
tion: thus the xcomponent of the force on a particle is equal to the xcomponent
of the rate of change of momentum of that particle, or fat = g; (mvz), (10.5) and similarly for the y and zdirections. Therefore Eq. (10.3) is really three
equations, one for each direction. In addition to the law of conservation of momentum, there is another inter
esting consequence of Newton’s Second Law, to be proved later, but merely stated
nOW. This principle is that the laws of phy51cs will look the same whether we are
standing still or moving with a uniform speed in a straight line. For example, a
child bouncing a ball in an airplane ﬁnds that the ball bounces the same as though
he were bouncing it on the ground. Even though the airplane is moving with a
very high veloc1ty, unless it changes its velocity, the laws look the same to the
child as they do when the airplane is standing still. This is the socalled relativity
principle. As we use it here we shall call it “Galilean relativity” to distinguish it
from the more careful analysis made by Einstein, which we shall study later. We have just derived the law of conservation of momentum from Newton’s
laws, and we could go on from here to ﬁnd the special laws that describe impacts
and collisions. But for the sake of variety, and also as an illustration of a kind of
reasoning that can be used in physics in other Circumstances where, for example,
one might not know Newton's laws and might take a different approach, we shall
discuss the laws of impacts and collisions from a completely diﬁerent point of
view. We shall base our discussion on the principle of Galilean relativity, stated
above, and shall end up with the law of conservatlon of momentum. We shall start by assuming that nature would look the same if we run along
at a certain speed and watch it as it would if we were standing still. Before dis 103 cussing collisions in which two bodies collide and stick together, or come together
and bounce apart, we shall ﬁrst consider two bod1es that are held together by a
spring or something else, and are then suddenly released and pushed by the spring
or perhaps by a little explosion. Further, we shall consider motion in only one
direction. First, let us suppose that the two objects are exactly the same, are nice
symmetrical objects, and then we have a little explosion between them. After the
explos1on, one of the bodies will be movmg, let us say toward the right, With a
velocity 1/. Then it appears reasonable that the other body is moving toward the
left with a velocity I), because if the objects are alike there is no reason for right or
left to be preferred and so the bodies would do something that is symmetrical. This
is an illustration of a kind of thinkmg that is very useful in many problems but
would not be brought out if we just started with the formulas. The ﬁrst result from our experiment is that equal objects will have equal
speed, but now suppose that we have two objects made of different materials,
say copper and aluminum, and we make the two masses equal. We shall now
suppose that if we do the experiment with two masses that are equal, even though
the objects are not ident1cal, the veloc1ties will be equal. Someone might object:
“But you knOW, you could do it backwards, you did not have to suppose that.
You could deﬁne equal masses to mean two masses that acquire equal velocities
in this experiment.” We follow that suggestion and make a little explosion between
the copper and a very large plece of aluminum, so heavy that the copper ﬁles out
and the aluminum hardly budges. That is too much aluminum, so we reduce the
amount until there is Just a very tiny piece, then when we make the explosion the
aluminum goes ﬂying away, and the copper hardly budges. That is not enough alu
minum. Evrdently there is some right amount in between; so we keep adjusting
the amount until the velocities come out equal. Very well then—let us turn it
around, and say that when the velocities are equal, the masses are equal. This
appears to be Just a deﬁnition, and it seems remarkable that we can transform
physical laws Into mere deﬁnitions. Nevertheless, there are some physical laws
involved, and if we accept this deﬁnitlon of equal masses, we immediately ﬁnd one
of the laws, as follows. Suppose we know from the foregoing experiment that two pieces of matter,
A and B (of copper and aluminum), have equal masses, and we compare a third
body, say a piece of gold, with the copper in the same manner as above, making
sure that its mass is equal to the mass of the copper. If we now make the experiment
between the aluminum and the gold, there is nothing in logic that says these masses
must be equal; however, the experzment shows that they actually are. So now, by
experiment, we have found a new law. A statement of this law might be: If two
masses are each equal to a third mass (as determined by equal velocities in this
experiment), then they are equal to each other. (This statement does not follow
at all from a similar statement used as a postulate regarding mathematical quanti
ties.) From this example we can see how quickly we start to infer things if we are
careless. It is not just a deﬁnition to say the masses are equal when the velocities
are equal, because to say the masses are equal is to imply the mathematical laws
of equality, which in turn makes a prediction about an experiment. As a second example, suppose that A and B are found to be equal by doing
the experiment with one strength of explosion, which gives a certain velocity; if
we then use a stronger explosion, W111 it be true or not true that the velocities now
obtained are equal? Again, in logic there is nothing that can decide th1s question,
but experiment shows that it is true. So, here is another law, which might be
stated: If two bodies have equal masses, as measured by equal velocities at one
velocity, they W111 have equal masses when measured at another velocity. From
these examples we see that what appeared to be only a deﬁnition really involved
some laws of physics. In the development that follows we shall assume it is true that equal masses
have equal and opposite velocities when an explosion occurs between them. We
shall make another assumption in the inverse case: If two identical objects, moving
in opposite directions with equal veloc1ties, collide and stick together by some klnd
of glue, then which way will they be moving after the collision? This is again a 10—4 symmetrical situation, with no preference between right and left, so we assume
that they stand still. We shall also suppose that any two objects of equal mass,
even if the objects are made of different materials, which collide and stick together, when moving with the same velocity in opposite directions will come to rest after
the collision. 103 Momentum is conserved! We can verify the above assumptions experimentally: ﬁrst, that if two station
ary objects of equal mass are separated by an explosion they will move apart with
the same speed, and second, if two objects of equal mass, coming together with the
same speed, collide and stick together they wrll stop. This we can do by means of
a marvelous invention called an air trough,* which gets rid of friction, the thing
which continually bothered Galileo (Fig. 10—1). He could not do experiments by
sliding things because they do not slide freely, but, by adding a magic touch, we
can today get rid of friction. Our objects will slide without difﬁculty, on and on at
a constant velocity, as advertised by Galileo. This is done by supporting the objects
on air. Because air has very low friction, an object glides along with practically
constant velocity when there is no applied force. First, we use two glide blocks
which have been made carefully to have the same weight, or mass (their weight
was measured really, but we know that this weight is proportional to the mass),
and we place a small explosive cap in a closed cylinder between the two blocks
(Fig. 10—2). We shall start the blocks from rest at the center point of the track and
force them apart by exploding the cap with an electric spark. What should happen?
If the speeds are equal when they ﬂy apart, they should arrive at the ends of the
trough at the same time. On reaching the ends they will both bounce back with
practically opposite velocity, and will come together and stop at the center where
they started. It is a good test; when it is actually done the result is just as we
have described (Fig. 10—3). Now the next thing we would like to ﬁgure out is what happens in a less simple
situation. Suppose we have two equal masses, one moving with velocity v and the
other standing still, and they collide and stick; what is going to happen? There
is a mass 2m altogether when we are ﬁnished, drifting with an unknown velocity.
What velocity? That is the problem. To ﬁnd the answer, we make the assumption
that if we ride along in a car, physics will look the same as if we are standing still.
We start with the knowledge that two equal masses, moving in opposite directions
with equal speeds 2), will stop dead when they collide. Now suppose that while
this happens, we are riding by in an automobile, at a velocity — 22. Then what does
it look like? Since we are riding along with one of the two masses which are coming
together, that one appears to us to have zero velocity. The other mass, however,
going the other way with velocity 12, will appear to be coming toward us at a velocity
22) (Fig. 10—4). Finally, the combined masses after collision will seem to be passing
by with velocity 1). We therefore conclude that an object with velocity 2v, hitting
an equal one at rest, will end up with velocity v, or what is mathematically exactly
the same, an object with velocity v hitting and sticking to one at rest will produce
an object moving with velocity 22/2. Note that if we multiply the mass and the
velocity beforehand and add them together, mv + 0, we get the same answer as
when we multiply the mass and the velocity of everything afterwards, 2m times
v/2. So that tells us what happens when a mass of velocity 2) hits one standing still. In exactly the same manner we can deduce what happens when equal objects
having any two velocities hit each other. Suppose we have two equal bodies with velocities DI and 02, respectively,
which collide and stick together. What is their velocity 1) after the collision?
Again we ride by in an automobile, say at velocity v2, so that one body appears to
be at rest. The other then appears to have a velocity v1 — 122, and we have the
same case that we had before. When it is all ﬁnished they will be moving at
%(2)1 — 02) with respect to the car. What then is the actual speed on the ground? * H. V. Neher and R. B. Leighton, Amer. Jour. of Phys. 3], 255 (1963).
10—5 SMALL HOLES
(JETS) COMPRESSED
AIR SUPPLY BUMPER SPRING TOY PISTOL CAP SPARK ELECTRODE SW“ <7 7”: CYLINDER PISTON BUIVPER SPRING Fig. 10—2. Sectional view of gliders
with explosive interaction cylinder attach
ment. Fig. 10—3. Schematic view of action
reaction experiment with equal masses. VIEW FROM VIEW FROM
CENTER OF MASS MOVING CAR
(CAR VELOCITY I V) V —O 4V 2V> 0 IE] [E] BEFORECOLLISION E] LE] V=° v—> m AFTER COLLISION m Fig. 10—4. Two views of an inelastic
collision between equal masses. VIEW FROM "LAB" VIEW FROM CAR vip vap W E [E] serene cocusaou [liq Ciiij v—> vaivi— H
m AFTER COLLISION Fig. 10—5. Two views of another
inelastic collision between equal masses. ZD+A——> D—> Fig. 10—6. An experiment to verify
that a mass m with velocity v striking a
mass m with zero velocity gives 2m with
velocity v/2. VIEW FROM VIEW FROM
CM SYSTEM CAR V V/2 SW? 0
<— E} [E] BEFORE emuson é [:rﬂ 0 VIZ—O EFLj AFTER cowscon [E Fig. 10‘7. Two views of an inelastic
collision between m and 2m. 0 0\I/o o o
;;
o ev v —> o 0
Ci] [3“] El IE [El
4w2 V/2 o 0
[E13 Eli] [El
out/I2 v13 +
DIE] m
Fig. 10—8. Action and reaction be tween 2m and 3m. It is v = an — 222) + 02 or %(ul + 222) (Fig. 10—5). Again we note that mvl + mm = 2m(vl + vz)/2. (10.6) Thus, using this principle, we can analyze any kind of collision in which two
bodies of equal mass hit each other and stick. In fact, although we have worked
only in one dimension, we can ﬁnd out a great deal about much more complicated
collisions by imagining that we are riding by in a car in some oblique direction.
The principle is the same, but the details get somewhat complicated. In order to test experimentally whether an object movmg with velocity v,
colliding With an equal one at rest, forms an object moving with velocity 12/2, we
may perform the following experiment with our airtrough apparatus. We place
in the trough three equally massive objects, two of which are initially joined to
gether with our exploswe cylinder deVice, the third being very near to but slightly
separated from these and provided with a sticky bumper so that it Will stick to
another object which hits it. Now, a moment after the explosion, we have two
objects of mass m moving with equal and opposite velocities v. A moment after
that, one of these collides with the third object and makes an object of mass 2m
moving, so we believe, With veloc1ty v/Z. How do we test whether it is really v/2?
By arranging the initial positions of the masses on the trough so that the distances
to the ends are not equal, but are in the ratio 2:1. Thus our ﬁrst mass, which
continues to move With veloc1ty 2;, should cover twice as much distance in a given
time as the two which are stuck together (allowmg for the small distance travelled
by the second object before it collided with the third). The mass m and the mass
2m should reach the ends at the same time, and when we try it, we ﬁnd that they
do (Fig. 10—6). The next problem that we want to work out is what happens if we have two
different masses. Let us take a mass m and a mass 2m and apply our explosive
interaction. What Will happen then? If, as a result of the explosion, in moves with
velocity 1), With what velocity does 2m move? The experiment we have just done
may be repeated with zero separation between the second and third masses. and
when we try it we get the same result, namely, the reacting masses m and 2m
attain velocities —v and u/Z. Thus the direct reaction between m and 2m gives
the same result as the symmetrical reaction between m and m, followed by a colliSion
between m and a third mass m in which they stick together. Furthermore, we ﬁnd
that the masses m and 2m returning from the ends of the trough, with their veloci
ties (nearly) exactly reversed, stop dead if they stick together. Now the next question we may ask is this. What will happen if a mass m with
velocity u, say, hits and sticks to another mass 2m at rest? This is very easy to
answer using our principle of Galilean relativity, for we simply watch the collision
which we have just described from a car moving with velocity —v/2 (Fig. 10—7).
From the car, the velocuies are 12’] = v — 22(car) = v + v/Z = 30/2
and
PL; = —v/2 — 2'(car) = —0/2 + 0/2 = 0. After the collision, the mass 3m appears to us to be moving with velocity v/2.
Thus we have the answer, i.e., the ratio of veIOCities before and after collision is
3 to 1: if an object of mass m collides with a stationary object of mass 2m, then the
whole thing moves off, stuck together, with a velocity 1/3 as much. The general
rule again is that the sum of the products of the masses and the velocities stays the
same: miy + 0 equals 3m times tr/3, so we are gradually building up the theorem
of the conservation of momentum, piece by piece. Now we have one against two. Using the same arguments, we can predict the
result of one against three. two against three, etc. The case of two against three,
starting from rest, is shown in Fig. 10—8. In every case we ﬁnd that the mass of the ﬁrst object times its velocity, plus
the mass of the second object times its velocity, is equal to the total mass of the
ﬁnal object times its veloc1ty. These are all examples, then, of the conservation of 1 0—6 momentum. Starting from simple, symmetrical cases, we have demonstrated the
law for more complex cases. We could, in fact, do it for any rational mass ratio,
and since every ratio is exceedingly close to a rational ratio, we can handle every
ratio as precisely as we wish. 10—4 Momentum and energy All the foregoing examples are simple cases where the bodies collide and stick
together, or were initially stuck together and later separated by an explosion.
However, there are situations in which the bodies do not cohere, as, for example,
two bodies of equal mass which collide with equal speeds and then rebound.
For a brief moment they are in contact and both are compressed. At the instant
of maximum compression they both have zero velocity and energy is stored in the
elastic bodies, as in a compressed spring. This energy is derived from the kinetic
energy the bodies had before the collision, which becomes zero at the instant their
velocity is zero. The loss of kinetic energy is only momentary, however. The
compressed condition is analogous to the cap that releases energy in an explosion.
The bodies are immediately decompressed in a kind of explosion, and ﬂy apart
again; but we already know that case—the bodies ﬂy apart with equal speeds.
However, this speed of rebound is less, in general, than the initial speed, because
not all the energy is available for the explosion, depending on the material. If the
material is putty no kinetic energy is recovered, but if it is something more rigid,
some kinetic energy is usually regained. In the collision the rest of the kinetic
energy is transformed into heat and vibrational energy—the bodies are hot and
vibrating. The vibrational energy also is soon transformed into heat. It is possible
to make the colliding bodies from highly elastic materials, such as steel, with
carefully designed spring bumpers, so that the collision generates very little heat
and Vibration. In these circumstances the velocities of rebound are practically
equal to the initial velocities; such a collision is called elastic. That the velocities before and after an elastic collision are equal is not a matter
of conservation of momentum, but a matter of conservation of kinetic energy.
That the speeds of the bodies rebounding after a symmetrical collision are equal
to each other, however, is a matter of conservation of momentum. We might similarly analyze collisions between bodies of different masses,
different initial velocities, and various degrees of elasticity, and determine the ﬁnal
velocities and the loss of kinetic energy, but we shall not go into the details of
these processes. Elastic collisions are especially interesting for systems that have no internal
“gears, wheels, or parts.” Then when there is a collision there is nowhere for the
energy to be impounded, because the objects that move apart are in the same
condition as when they collided. Therefore, between very elementary objects, the
collisions are always elastic or very nearly elastic. For instance, the collisions
between atoms or molecules in a gas are said to be perfectly elastic. Although this
is an excellent approximation, even such collisions are not perfectly elastic; other
wise one could not understand how energy in the form of light or heat radiation
could come out of a gas. Once in a while, in a gas collision, a lowenergy infrared
ray is emitted, but this occurrence is very rare and the energy emitted is very small.
So, for most purposes, collisions of molecules in gases are considered to be per
fectly elastic. As an interesting example, let us consider an elaszic collision between two
objects of equal mass. If they come together with the same speed, they would
come apart at that same speed, by symmetry. But now look at this in another
circumstance, in which one of them is moving with velocity v and the other one is
at reSt. What happens? We have been through thls before. We watch the sym
metrical collision from a car moving along with one of the objects, and we ﬁnd
that if a stationary body is struck elastically by another body of exactly the same
mass, the moving body stops, and the one that was standing still now moves away
with the same speed that the other one had; the bodies simply exchange velocities.
This behavior can easily be demonstrated with a suitable impact apparatus. More 10—7 generally, if both bodies are moving, with different velocities, they simply exchange
velocity at impact. Another example of an almost elastic interaction is magnetism. If we arrange
a pair of Ushaped magnets in our glide blocks, so that they repel each other,
when one drifts quietly up to the other, it pushes it away and stands perfectly still,
and now the other goes along, frictionlessly. The principle of conservation of momentum is very useful, because it enables
us to solve many problems without knowing the details. We did not know the
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, M, ejects a small piece, of mass m, with
a terriﬁc velocity V relative to the rocket. After this the rocket, if it were originally
standing still, will be moving with a small velocity, 2). Using the principle of con
servation of momentum, we can calculate this velocity to be m U — 11—! V.
So long as material is being ejected, the rocket continues to pick up speed.
Rocket propulsion is essentially the same as the recoil of a gun: there is no need for any air to push against. 105 Relativistic momentum In modern times the law of conservation of momentum has undergone certain
modiﬁcations. However, the law is still true today, the modiﬁcations being mainly
in the deﬁnitions of things. In the theory of relativity it turns out that we do have
conservation of momentum; the particles have mass and the momentum is still
given by my, the mass times the velocity, but the mass changes with the velocity, hence the momentum also changes. The mass varies with velocity according to the law
mo V1 — vZ/c2 ’ where m0 is the mass of the body at rest and c is the speed of light. It is easy to
see from the formula that there is negligible diﬁerence between m and mo unless v 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 m: (10.7) m 003; m 001/ m 01) z pI=Vl—v2/c2’ py=\/l~1)2/c2’ pz=\/1—1)2/c2’ where 02 = 22.2, + v: + v3. If the xcomponents are summed over all the inter
acting particles, both before and after a collision, the sums are equal; that is,
momentum is conserved in the xdirection. 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 different 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 momen
tum?” The 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,
if the squares of the velocities are summed. This 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 of energy in the form of heat is not very obvious.
On the other hand, if one sums the velocities, which have direction, and ﬁnds 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 108 (10.8) when it moves as a whole. Therefore momentum, as a mechanical quantity, is
difﬁcult to hide. Nevertheless, momentum can be hidden—in the electromagnetic
ﬁeld, for example. This case is another effect of relativity. One of the propositions of Newton was that interactions at a distance are
instantaneous. It turns out that such is not the case; in situations involving
electrical forces, for instance, if an electrical charge at one location is suddenly
moved, the effects on another charge, at another place, do not appear instantane
ously—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 pick up some momentum, but the second charge has felt nothing and
has not yet changed its momentum. It takes time for the inﬂuence 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. Of course after the second charge
has felt the effect of the ﬁrst one and all is quieted down, the momentum equation
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 mo
mentum besides that of the particle, my, and that is momentum in the electro
magnetic ﬁeld. If we add the ﬁeld momentum to the momentum of the particles,
then momentum is conserved at any moment all the time. The fact that the electro
magnetic ﬁeld can possess momentum and energy makes that ﬁeld very real, and
so, for better understanding, the original idea that there are just the forces between
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 ﬁeld has waves, which we call light; it turns out that light also
carries momentum with it, so when light impinges on an object it 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 is 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 different thing—
it is no longer my. It is hard to deﬁne exactly what is meant by the velocity of a
particle, but momentum still exists. In quantum mechanics the difference is that
when the particles are represented as particles, the momentum is still my, 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 momen
tum. In spite of the differences, the law of conservation of momentum holds also
in quantum mechanics. Even though the law f = ma is false, and all the deriva
tions of Newton were wrong for the conservation of momentum, in quantum
mechanics, nevertheless, in the end, that particular law maintains itself! 10—9 ...
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 Spring '09
 LeeKinohara
 Physics, Momentum

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