Collisions and conservation laws
About this free course
This free course is an adapted extract from the Open University course S217 Physics: from classical to
quantum www.open.ac.uk/courses/modules/s217.
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Contents
Introduction
Learning Outcomes
1 Analysing collisions
2 Elastic and inelastic collisions
3 Elastic collisions in one dimension
3.1 Elastic collisions with a stationary target
3.2 Elastic collisions in general
3.3 Four special cases of general elastic collisions
4 Elastic collisions in two or three dimensions
5 Inelastic collisions
6 Collisions all around us
6.1 Collisions in space
6.2 Collisions on the roads
6.3 Collisions on a small-scale
7 Relativistic collisions
7.1 Relativistic momentum
7.2 Relativistic kinetic energy
Conclusion
Keep on learning
Glossary
Acknowledgements
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Introduction
Introduction
This free course, Collisions and conservation laws, is about collisions and how they may
be understood using concepts referred to as the conservation of linear momentum and
the conservation of kinetic energy. We’ll begin by defining some important quantities
that will be used in what follows. A collision is a brief, but often powerful, interaction
between two bodies in close proximity; we often idealise the situation in physics problems
to consider collisions of pointlike objects travelling along a line or in a plane. Linear
momentum is a physical property of a body in motion which is equivalent to its mass
multiplied by its velocity. It is a vector quantity so possesses both a magnitude and a
direction, which is the same as the direction of the body’s velocity. The kinetic energy of
a body is a measure of the energy it possesses by virtue of its motion. It is a scalar
quantity, possessing a magnitude only, which is equivalent to half the body’s mass
multiplied by the square of its speed.
This OpenLearn course is an adapted extract from the Open University course
S217 Physics: from classical to quantum.
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Learning Outcomes
After studying this course, you should be able to:
l
understand the meaning of all the newly defined (emboldened) terms introduced
l
describe the essential features of elastic and inelastic collisions, and give examples of each
l
use the law of conservation of momentum, and (when appropriate) the law of conservation of kinetic energy, to
solve a variety of simple collision problems.
1 Analysing collisions
1 Analysing collisions
The analysis of collisions is of fundamental importance in physics, particularly in nuclear
and particle physics, and the techniques used to analyse collisions are well established
and widely used. They are also very firmly rooted in the basic conservation principles (or
‘conservation laws’ as they are sometimes known), particularly those of momentum and
energy.
2 Elastic and inelastic collisions
When starting to investigate collision problems, we usually consider situations that either
start or end with a single body. The reason for this self-imposed limitation is that such
problems can be solved by applying momentum conservation alone, namely the result
that the total linear momentum of an isolated system is constant. The analysis of more
general collisions requires the use of other principles in addition to momentum
conservation. To illustrate this, we now consider a one-dimensional problem in which two
colliding bodies with known masses and , and with known initial velocities and collide and
then separate with final velocities and . The problem is that of finding the two unknowns
and . Conservation of momentum in the -direction provides only one equation linking
these two unknowns:
(1)
which is insufficient to determine both unknowns.
In the absence of any detailed knowledge about the forces involved in the collision, the
usual source of an additional relationship between and comes from some consideration of
the translational kinetic energy involved. The precise form of this additional relationship
depends on the nature of the collision.
Collisions may be classified by comparing the total (translational) kinetic energy of the
colliding bodies before and after the collision. If there is no change in the total kinetic
energy, then the collision is an elastic collision. If the kinetic energy after the collision is
less than that before the collision then the collision is an inelastic collision. In some
situations (e.g. where internal potential energy is released) the total kinetic energy may
even increase in the collision; in which case the collision is said to be a superelastic
collision.
In the simplest case, when the collision is elastic, the consequent conservation of kinetic
energy means that
(2)
This equation, together with Equation 1 will allow and to be determined provided the
masses and initial velocities have been specified. We consider this situation in more detail
in the next section.
Real collisions between macroscopic objects are usually inelastic but some collisions,
such as those between steel ball bearings or between billiard balls, are very nearly elastic.
Collisions between subatomic particles, such as electrons and/or protons, commonly are
elastic. The kinetic energy which is lost in an inelastic collision appears as energy of a
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3 Elastic collisions in one dimension
different form (e.g. thermal energy, sound energy, light energy, etc.), so that the total
energy is conserved. Collisions in which the bodies stick together on collision and move
off together afterwards, are examples of completely inelastic collisions. In these cases
the maximum amount of kinetic energy, consistent with momentum conservation, is lost.
(Momentum conservation usually implies that the final body or bodies must be moving
and this inevitably implies that there must be some final kinetic energy; it is the remainder
of the initial kinetic energy, after this final kinetic energy has been subtracted, that is lost in
a completely inelastic collision.)
3 Elastic collisions in one dimension
In this section you will examine the outcomes of various elastic collisions in one
dimension. These are essentially particular cases of the general elastic collision
described by Equation 1 and Equation 2. Cataloguing one-dimensional elastic collisions
may sound like a rather esoteric pastime, but as you will see, you have probably
witnessed many collisions of this type, and may even have paid handsomely for the
privilege.
3.1 Elastic collisions with a stationary target
We begin with an example of a one-dimensional elastic collision between two particles of
identical mass, one of which is initially stationary. Our aim now is to find the final velocity
of each particle after the collision.
Activity 1
A particle of mass moves along the -axis with velocity and collides elastically with an
identical particle at rest. What are the velocities of the two particles after the collision?
Answer
Let the final velocities be and . Conservation of momentum along the -axis gives
(3)
and conservation of kinetic energy for this elastic collision gives
(4)
By eliminating common factors, Equation 3 can be simplified to give
(5)
and Equation 4 can be treated similarly to give
(6)
Rearranging Equation 6 gives
the right-hand side of which may be rewritten using the general identity , thus
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3 Elastic collisions in one dimension
Dividing both sides of this last equation by ,
and using Equation 5 to simplify the resulting right-hand side gives
Comparing this expression for with that in Equation 5 shows that
The result of Example 1 will be familiar to anyone who has seen the head-on collision of
two bowls on a bowling green. The moving one stops, and the one that was initially
stationary moves off with the original velocity of the first. In effect, the bowls exchange
velocities.
Activity 2
Predict qualitatively (i.e. without calculation) what would happen when a body of mass
collides with another body of mass that is initially at rest if:
(a)
(The symbol should be read as ‘is very much greater than’.)
Answer
Experience should tell you that a high-mass projectile fired at a low-mass target would
be essentially unaffected by the collision.
(b)
.
Answer
A low-mass projectile fired at a massive target would bounce back with unchanged
speed.
3.2 Elastic collisions in general
We now consider the general one-dimensional elastic collision between particles of mass
and which move with initial velocities and before the collision and final velocities and after
the collision. As stated earlier, the outcome of collisions of this kind is determined by
Equation 1 and Equation 2. We shall not write down the details (though you might like to
work them out for yourself) but by arguments similar to those used in Example 1, the
following result can be obtained.
If the initial velocity of particle relative to particle is taken to be
and if the final velocity of particle relative to particle is taken to be
then, as a result of an elastic collision
In other words:
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In a one-dimensional elastic collision between two particles the relative velocity of approach
is the negative of the relative velocity of separation
(7)
Combining this result (which incorporates the conservation of kinetic energy) with
Equation 1 (which expresses conservation of momentum), leads to the following
expressions for the final velocities:
(8)
(9)
In the next two exercises you can use Equations 8 and 9. However, these equations are
complicated so you are not expected to memorise them. You should be able to solve this
type of question starting from the equations of conservation of momentum (Equation 1)
and kinetic energy (Equation 2).
Activity 3
A neutron of mass rebounds elastically in a head-on collision with a gold nucleus of
mass that is initially at rest. What fraction of the neutron’s initial kinetic energy is
transferred to the recoiling gold nucleus? Repeat this calculation when the target is a
carbon nucleus at rest and of mass .
Answer
The initial kinetic energy is
and the final kinetic energy is
therefore the loss in energy is
and the fractional loss is
With in Equation 8,
we can write the fractional loss as
For gold
For carbon
So, a low-mass nucleus is much more effective than a more massive nucleus when it
comes to slowing down fast neutrons by elastic collisions. It is because of this fact that
carbon is used in a nuclear reactor for just this purpose.
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Activity 4
A tennis player returns a service in the direction of the server. The ball of mass arrives
at the racket of mass with a speed of and the racket is travelling at at impact. Calculate
the velocity of the returning ball, assuming elastic conditions.
Answer
We designate the ball as particle and the racket as particle , with the ball initially
travelling along the positive -direction. From Equation 8
so
i.e.
3.3 Four special cases of general elastic collisions
It is interesting to examine these results for and in a few special cases, including some
that have been mentioned earlier. The cases are illustrated in Animation 1, and have
many familiar sporting applications.
Interactive content is not available in this format.
Animation 1 Animation showing body with mass and initial speed moving in one
dimension and colliding with body with mass and speed initially moving in the opposite
direction. The controls in the animation allow you to change the values of , , and . Press
the play button to see what happens during the elastic collision.
Activity 5
Using Animation 1, explore what happens in the cases listed below and find, using
Equation 8 and Equation 9, expressions for and :
(a)
Answer
With the masses equal, we can see that if particle is moving slowly and approaching
particle which is moving quickly, then after the collision, the particles rebound but now
particle is moving quickly and particle is moving slowly. In fact, Equation 8 and
Equation 9 give and so the velocities of the particles are simply exchanged.
(b)
Answer
The motion of the high-mass particle is virtually unchanged by the collision. If is very
small compared with , then and . With , Equation 8 and Equation 9 then give and . The
low-mass particle moves off with a velocity of twice that of the high-mass particle.
(c)
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Answer
The low-mass particle rebounds with almost unchanged speed while the high-mass
particle remains essentially at rest. If is very small compared with (which is stationary),
then Equation 8 and Equation 9 lead to and .
(d)
Answer
The low-mass particle bounces back with higher speed, while the high-mass particle
continues essentially unaffected by the collision. If is negligible compared with , and
the two bodies approach head-on with equal speeds then Equation 8 and Equation 9
lead to and .
The results for the four special cases in the exercises accord with common experience.
Let’s summarise them:
1
: the particles simply exchange velocities. We saw this result earlier in this section; it
is a familiar occurrence in bowls and snooker.
2
; : the low-mass particle moves off with a velocity of twice that of the high-mass
particle. Tennis players will be familiar with this case from serving.
3
; : the low-mass particle rebounds with almost unchanged speed while the highmass particle remains essentially at rest. Golfers whose ball hits a tree will recognise
this situation.
4
; : the low-mass particle bounces back with three times the initial speed, while the
high-mass particle continues essentially unaffected by the collision. This case will be
recognised by a batsman playing cricket or by a tennis player returning a serve.
The results quoted above under points 2, 3 and 4 give an upper limit to the speed that can
be imparted to a ball hit by a club, bat or racket.
Activity 6
The ‘Newton’s cradle’ executive toy shown below performs repeated collisions
between one ball and a row of four identical balls. You may assume the collisions are
perfectly elastic.
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Figure 1 ‘Newton’s cradle’ executive toy: a row of four balls suspended from a frame
collide in sequence
(a) Explain how this toy fits into the framework of four special cases enumerated
above.
Answer
While at first glance, we have the situation ; , most closely corresponding to point 3
above, the behaviour of the toy is not as predicted by point 3. In fact, the toy executes
a rapid series of four repeated collisions corresponding to point 1 above: with the case
. The second ball moves instantaneously with velocity , colliding immediately with the
third ball, which moves instantaneously with velocity , and so on until the fifth ball
carries the momentum away with velocity .
(b) Gravitational potential energy is energy possessed by an object by virtue of its
height – objects further from the surface of the Earth will have a greater gravitational
potential energy. As an object falls, its gravitational potential energy will be converted
into kinetic energy, and as an object rises, its kinetic energy will be converted into
gravitational potential energy. What role does the exchange of gravitational potential
energy and kinetic energy play in the dynamics of the first and fifth balls?
Answer
After the four collisions, the fifth ball moves in a pendulum-like trajectory until its kinetic
energy is converted to gravitational potential energy. It comes instantaneously to rest,
then moves back to collide with the fourth ball with velocity (neglecting air resistance
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4 Elastic collisions in two or three dimensions
encountered during the motion). The same thing happens when the first ball is set in
motion by the second ball, it begins to move as a pendulum with velocity . This kinetic
energy is again converted to gravitational energy until the first ball comes
instantaneously to rest, then moves back to collide with the second ball with velocity ,
repeating the cycle.
4 Elastic collisions in two or three
dimensions
The laws of conservation of momentum and energy that we used to analyse elastic
collisions in one dimension are also used to analyse elastic collisions in two or three
dimensions. We simply treat the motions in each dimension as independent, and apply
conservation of momentum separately along each Cartesian coordinate axis. Kinetic
energy conservation continues to provide one additional equation relating the squares of
the particle speeds. Since we have been careful to use vector notation throughout, this
extension to two or three dimensions is easily made.
Consider the elastic collision of two identical bodies of mass , one at rest and the other
approaching with velocity . The particles are no longer confined to move in one dimension,
so our -component equation (Equation 1), embodying conservation of momentum,
becomes a full vector equation:
The law of conservation of energy (Equation 2) does not change, so
These can be simplified to:
(10)
and
(11)
These equations are most easily interpreted by a diagram. Figure 2 shows how the three
vectors , and are related to one another. Equation 10 tells us that all three velocity vectors
must lie in a single plane, and that they must form a closed triangle. Equation 11 tells us
that the triangle must be a right-angled triangle, since its sides obey Pythagoras’ theorem.
The implication of this is striking, it means that the angle between and must be .
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Figure 2 (a) Elastic collision between particles of equal mass, with one at rest; (b) the
corresponding vector triangle.
Following the elastic collision of two identical particles, one of which is initially at rest, the
final velocities of the two particles will be at right-angles.
This is a simplifying feature of equal-mass collisions in two or three dimensions,
analogous to the simple result of the exchange of velocities, which we found in one
dimension.
You may have noticed that this result does not tell us exactly where the bodies go after the
collision. Any pair of final velocities which can be represented by Figure 2 will be equally
satisfactory, and there are an infinite number of these. The reason for this is that we have
said nothing about the shape or size of the bodies, or just how they collide. We usually
need to have additional information of this kind if we are to determine unique final
velocities in such cases. Figure 3 shows the outcome of a particular collision in which
spherical bodies make contact at a specific point. The location of this point is the sort of
additional information required to determine unique values for and .
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4 Elastic collisions in two or three dimensions
Figure 3 When ball strikes ball , the reaction forces at the contact point ensure that ball is
propelled away along the line of centres, as in snooker.
Activity 7
For the case illustrated in Figure 2 (two bodies of equal mass, one of which is initially at
rest), if the moving body has an initial speed of , and is deflected through in the
collision, find the magnitudes and directions of the velocities and .
Answer
We draw a vector triangle like the one shown in Figure 2b
Figure 4 Vector triangle
We can now see that
and is at the given angle of to the -axis; has a magnitude
and must be at to the -axis so that the two angles add up to .
Activity 8
In the same situation (Figure 2), if, instead of the outcome specified in Activity 7, the
speed of the moving body is reduced from to by the collision, find the final velocities.
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5 Inelastic collisions
Answer
Using the triangle in Figure 2 Pythagoras’ theorem tells us that , so
so that .
Now is at an angle and at an angle to the -axis. You will observe that the two angles
add up to , as they should.
When the masses of the two colliding particles are unequal the algebraic manipulations
required to solve elastic collision problems become rather complicated, but no new
physics is involved in the solution so we will not pursue such problems here.
5 Inelastic collisions
We now extend our discussion to include inelastic cases, where the total kinetic energy
changes during the one-dimensional collision.
First we consider the case where the two particles stick together on impact; this is an
example of a completely inelastic collision, which occurs with the maximum loss of kinetic
energy consistent with conservation of momentum. As a simple example, suppose we
have two bodies of equal mass, with one initially at rest. If the initial velocity of the other is
and the initial momentum is , the final momentum must be the same so, since the mass
has been doubled, the final velocity is and the final kinetic energy is therefore
Since the initial kinetic energy was twice as large as this, it follows that half the original
kinetic energy has been lost (mainly as thermal energy), during the collision.
For the more general case where the colliding masses are unequal, but they stick together
at collision, we still have and so momentum conservation implies that
and provides a full solution of the problem (a value for ), without recourse to energy.
To complete the picture, let us mention the general case where two particles collide but
where the transfer of kinetic energy into other forms is less than that for the completely
inelastic case. This problem has no general solution without more information, such as the
fraction of kinetic energy converted. Such problems have solutions which lie between
those for the two extremes of elastic and completely inelastic collisions but they must be
tackled on an individual basis, using the general principles of conservation of momentum
and energy.
You will see that in all these calculations we have not needed to invoke the rather
complicated forces involved in the interaction of the two particles, but rather have been
able to solve the problems using only the principles of conservation of momentum and
energy. This is a great simplification and illustrates the power of using conservation
principles whenever possible.
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6 Collisions all around us
6 Collisions all around us
Collisions are occurring around us all the time and on all size scales, from the very largest
to the very smallest. In the following sections, you will listen to and watch various audios
and videos which describe how these collisions may be understood.
6.1 Collisions in space
Collisions in space are a frequent occurrence. In the following audio, we describe how
such collisions may be understood in terms of the concepts introduced in this course.
Figure 5 An artist’s impression of a large asteroid impacting on the Earth causing global
extinction of a number of species of plants and animals.
Audio content is not available in this format.
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7 Relativistic collisions
Audio 1 Collisions in space
6.2 Collisions on the roads
The growing number of cars on today’s roads makes it increasingly likely that each driver
will be involved in at least one collision during their lifetime, making this a matter of
personal interest for all of us (Video 1).
Video content is not available in this format.
Video 1 The physics of colliding cars.
6.3 Collisions on a small-scale
In case you think it’s a long time since you personally were involved in a collision, you
should be aware that even the air that you breathe has its properties regulated by the
innumerable collisions that occur every second between the molecules in the air. The air
pressure that helps to keep your lungs inflated and enables you to breathe is a result of
the rate at which momentum is transferred between the molecules in the air and lung
tissue.
Collisions continue to be of importance in nuclear physics, but they are even more
significant in subnuclear physics. Collision experiments, usually at very high energies, are
almost synonymous with the experimental investigation of elementary particles such as
protons, and their supposedly fundamental constituents, the quarks and gluons. These
investigations are carried out with the aid of purpose-built particle accelerators, such as
the ones at The European Centre for Particle Physics (CERN) (Video 2) or Brookhaven
National Laboratory in the USA. Sophisticated detectors allow the energies and momenta
of the emerging particles to be measured, aiding the identification of the particles and the
analysis of their behaviour. The results give an indication of the underlying structure of the
colliding particles, and have revealed the existence of forms of matter that would still be
unknown and possibly even unsuspected were it not for collision experiments. The most
recent of these results is the observation of a particle consistent with the Higgs boson
announced by CERN in July 2012.
Video content is not available in this format.
Video 2 A look inside CERN.
7 Relativistic collisions
The high-energy collision experiments carried out at CERN, Brookhaven National
Laboratory and other such facilities, involve particles that travel at speeds close to that of
light. Under such circumstances the definitions of momentum and translational kinetic
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7 Relativistic collisions
energy, that play such an important role in Newtonian mechanics, reveal certain
shortcomings. It is still the case that translational kinetic energy is conserved in an elastic
collision and that the momentum of an isolated system is always conserved, but the
Newtonian expressions
are now recognised as approximations, valid only at low speeds (i.e. at speeds much less
than the speed of light), to more complicated expressions that work at any speed, up to
the speed of light. The breakthrough that led to this realisation was the development of
Einstein’s special theory of relativity in 1905. Here we shall quote a few of its wellestablished results concerning momentum and energy.
7.1 Relativistic momentum
According to Einstein’s theory the relativistic momentum of a particle with mass and
velocity is given by
(12)
where is the speed of the particle and is the speed of light in a vacuum. The speed of light
in a vacuum, , plays an important role throughout special relativity. Among other things it
represents an upper limit to the speed of any particle.
Equation 12 implies that the momentum of a particle increases more rapidly with
increasing speed than the Newtonian relation ( ) predicts. This is shown in Figure 6, where
the behaviour of the Newtonian and relativistic definitions of momentum magnitude are
compared. You can see the good agreement at low speed, but you can also see the
increasing discrepancy as the speed increases. Note that the relativistic definition does
not extend beyond . This reflects the fact that in special relativity it is impossible to
accelerate a particle with mass to the speed of light, as you will soon see.
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Figure 6 The magnitude of the momentum of a particle of mass plotted against the
particle’s speed according to Newtonian mechanics and special relativity. The Newtonian
relation closely approximates that of relativity for values of that are small compared with
the speed of light, .
7.2 Relativistic kinetic energy
One of the most celebrated aspects of special relativity is Einstein’s discovery of mass
energy, the energy that a particle has by virtue of its mass. The mass energy of a particle
of mass (sometimes called the rest mass in this context) is given by
(13)
The mass energy is also known as rest energy.
The reason for mentioning this relation here is that it plays a part in determining the kinetic
energy of a particle. How is this? Well, according to special relativity the total energy
(including the mass energy) of a particle of mass travelling with speed is
(14)
Since this quantity is the sum of the translational kinetic energy and the mass energy of
the particle it follows that, according to the theory of relativity, the translational kinetic
energy of a particle of mass and speed is
(15)
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Unlikely as it may seem, this expression actually agrees very closely with the Newtonian
expression for translational kinetic energy when is small compared with . The relativistic
and Newtonian definitions of translational kinetic energy are compared in Figure 7. The
figure also indicates one reason why it is impossible to accelerate a particle to the speed
of light; doing so would require the transfer of an unlimited amount of energy to the
particle.
Figure 7 The translational kinetic energy of a particle of mass plotted against the particle’s
speed according to Newtonian physics and special relativity. The Newtonian relation
closely approximates that of relativity for values of that are small compared with the speed
of light, .
In analysing high-speed relativistic collisions, it is the relativistic expressions for
momentum and energy that must be used rather than their Newtonian counterparts. In an
elastic collision, all of the quantities we have just defined will be conserved:
l
momentum
l
mass energy
l
kinetic energy
l
total energy.
However, many high-energy collisions are actually inelastic, and in a high-energy inelastic
collision the only quantities that are certain to be conserved are the momentum and total
energy. In a general inelastic collision, neither kinetic energy nor mass energy is
necessarily conserved. This means that in an inelastic collision it is quite possible for
particles to be created or destroyed, thereby increasing or decreasing the mass energy.
However, the conservation of total energy means that any change in mass energy must
be accompanied by a compensating change in the kinetic energy. Thus particles may be
created, but only at the expense of kinetic energy.
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Conclusion
The need for kinetic energy in order to create particles explains why advances in particle
physics often require the construction of powerful new particle accelerators. Increasing
the kinetic energy of the colliding particles increases the mass of the particles that may be
created in the collision and thus opens up the possibility of creating previously
undiscovered forms of matter. Figure 8 shows the tracks of particles created in one such
‘ultra-relativistic’ collision.
Figure 8 Tracks of particles coming from an ‘ultra-relativistic’ inelastic collision at CERN.
This section has made much use of the phrase ‘high-speed collision’. The meaning of the
term ‘high speed’ obviously depends on context. However, if we simply take it to refer to
speeds that are sufficiently high that there is a clear discrepancy between the Newtonian
and relativistic values of kinetic energy and momentum, then we can say that high speed
means greater than about . This may not be obvious from the curves in Figure 6 and
Figure 7 because of the scale that has been used to draw them, but is the threshold used
by physicists.
Conclusion
Having completed this free course, Collisions and conservation laws, you should now be
able to state the law of conservation of momentum and describe the essential features of
elastic and inelastic collisions. You should also be able to use the law of conservation of
momentum and (when appropriate) the law of conservation of kinetic energy to solve a
variety of simple collision problems. In addition, you should recognise the expressions for
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Keep on learning
momentum and energy that arise in special relativity and explain their implications for the
creation of new particles in high-speed inelastic collisions at CERN.
This OpenLearn course is an adapted extract from the Open University course
S217 Physics: from classical to quantum.
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Glossary
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Glossary
collision
A brief interaction between two or more particles or bodies in close proximity.
completely inelastic collision
A collision in which the colliding bodies stick together, resulting in the maximum loss of
kinetic energy consistent with conservation of momentum.
conservation of kinetic energy
The principle that the total kinetic energy of any isolated system is constant.
conservation of linear momentum
The principle that the total linear momentum of any isolated system is constant.
elastic collision
A collision in which kinetic energy is conserved.
inelastic collision
A collision in which kinetic energy is not conserved.
internal force
In the context of a given system, an internal force is a force that acts within the system
and which has a reaction that also acts within the system.
isolated system
A system which cannot exchange matter or energy with its environment. In the context
of mechanics, an isolated system is one that is subject only to internal forces.
kinetic energy
The energy that a body possesses by virtue of its motion.
law of conservation of linear momentum
See conservation of linear momentum.
linear momentum
The momentum associated with the translational motion of a body. For a particle of
mass travelling with velocity , the linear momentum is .
mass energy
The energy that a body possesses by virtue of its mass, as given by , where is the
speed of light in a vacuum. The existence of mass energy is one of the many
implications of the special theory of relativity. The mass energy of a free particle is the
difference between its (total) relativistic energy and its relativistic translational kinetic
energy. Mass energy is also known as rest energy.
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Acknowledgements
momentum
A vector quantity, useful in various situations as a measure of a body's tendency to
continue in its existing state of rotational or translational motion.
principle of conservation of linear momentum
See conservation of linear momentum.
relativistic collision
A collision involving sufficiently high speeds that its analysis requires the use of the
relativistic relations for momentum and energy rather than their Newtonian counterparts. Relativistic collisions are often inelastic and are characterised by the creation of
new particles and an associated increase in mass energy (at the expense of kinetic
energy).
relativistic energy
According to the theory of special relativity the total energy (including the mass energy)
of a particle of mass travelling with speed is
relativistic kinetic energy
According to the theory of special relativity, the translational kinetic energy of a particle
of mass and speed is equal to its total relativistic energy minus its mass energy
relativistic momentum
The momentum of a body according to the special theory of relativity. For a particle of
(rest) mass , travelling with velocity , the relativistic momentum is At speeds which are
small compared with the speed of light, , this reduces to the Newtonian expression .
superelastic collision
A collision in which the kinetic energy increases, typically as a result of the release of
potential energy.
system
That part of the Universe which is the subject of an investigation.
Acknowledgements
This free course was written by Professor Andrew Norton.
Except for third party materials and otherwise stated (see terms and conditions), this
content is made available under a
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 Licence.
The material acknowledged below is Proprietary and used under licence (not subject to
Creative Commons Licence). Grateful acknowledgement is made to the following sources
for permission to reproduce material in this free course:
Images
Course image: D. M. Eigler, IBM Research Division.
Figure 1: DemonDeLuxe (Dominique Toussaint). This file is licensed under the
Creative Commons Attribution-Share Alike Licence
Figure 5: gl0ck33 / 123RF
Figure 8: courtesy © CERN
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Acknowledgements
Audio-visual
Animation 1: © The Open University
Audio 1: © The Open University
Video 1: (6.2): from: Mother of All Collisions (2000) by The BBC for The Open University ©
The Open University and its licensors
Video 2: (6.3): from: Big Bang Night, The Big Bang Machine, BBC4 3 September 2000 ©
The BBC
Every effort has been made to contact copyright owners. If any have been inadvertently
overlooked, the publishers will be pleased to make the necessary arrangements at the
first opportunity.
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