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Solution for (a)

For part (a):

  1. Identify the knowns: v = 0.990 c ; m = 9.11 × 10 −31 kg.
  2. Identify the unknown: K rel .
  3. Express the answer as an equation: K rel = ( γ 1 ) m c 2 with γ = 1 1 u 2 / c 2 .
  4. Do the calculation. First calculate γ . Keep extra digits because this is an intermediate calculation:
    γ = 1 1 u 2 c 2 = 1 1 ( 0.990 c ) 2 c 2 = 7.0888 .

    Now use this value to calculate the kinetic energy:
    K rel = ( γ 1 ) m c 2 = ( 7.0888 1 ) ( 9.11 × 10 −31 kg ) ( 3.00 × 10 8 m/s 2 ) = 4.9922 × 10 −13 J.
  5. Convert units:
    K rel = ( 4.9922 × 10 −13 J) ( 1 MeV 1.60 × 10 −13 J ) = 3.12 MeV.

Solution for (b)

For part (b):

  1. List the knowns: v = 0.990 c ; m = 9.11 × 10 −31 kg.
  2. List the unknown: K rel .
  3. Express the answer as an equation: K class = 1 2 m u 2 .
  4. Do the calculation:
    K class = 1 2 m u 2 = 1 2 ( 9.11 × 10 −31 kg ) ( 0.990 ) 2 ( 3.00 × 10 8 m/s ) 2 = 4.0179 × 10 −14 J.
  5. Convert units:
    K class = 4.0179 × 10 −14 J ( 1 MeV 1.60 × 10 −13 J ) = 0.251 Mev.

Significance

As might be expected, because the velocity is 99.0% of the speed of light, the classical kinetic energy differs significantly from the correct relativistic value. Note also that the classical value is much smaller than the relativistic value. In fact, K rel / K class = 12.4 in this case. This illustrates how difficult it is to get a mass moving close to the speed of light. Much more energy is needed than predicted classically. Ever-increasing amounts of energy are needed to get the velocity of a mass a little closer to that of light. An energy of 3 MeV is a very small amount for an electron, and it can be achieved with present-day particle accelerators. SLAC, for example, can accelerate electrons to over 50 × 10 9 eV = 50,000 MeV.

Is there any point in getting v a little closer to c than 99.0% or 99.9%? The answer is yes. We learn a great deal by doing this. The energy that goes into a high-velocity mass can be converted into any other form, including into entirely new particles. In the Large Hadron Collider in [link] , charged particles are accelerated before entering the ring-like structure. There, two beams of particles are accelerated to their final speed of about 99.7% the speed of light in opposite directions, and made to collide, producing totally new species of particles. Most of what we know about the substructure of matter and the collection of exotic short-lived particles in nature has been learned this way. Patterns in the characteristics of these previously unknown particles hint at a basic substructure for all matter. These particles and some of their characteristics will be discussed in a later chapter on particle physics.

A photo of Geneva with the location of CERN and the two rings are shown. The smaller ring is inside but tangent to the larger one.
The European Organization for Nuclear Research (called CERN after its French name) operates the largest particle accelerator in the world, straddling the border between France and Switzerland.

Total relativistic energy

The expression for kinetic energy can be rearranged to:

E = m u 2 1 u 2 / c 2 = K + m c 2 .

Einstein argued in a separate article, also later published in 1905, that if the energy of a particle changes by Δ E , its mass changes by Δ m = Δ E / c 2 . Abundant experimental evidence since then confirms that m c 2 corresponds to the energy that the particle of mass m has when at rest. For example, when a neutral pion of mass m at rest decays into two photons, the photons have zero mass but are observed to have total energy corresponding to m c 2 for the pion. Similarly, when a particle of mass m decays into two or more particles with smaller total mass, the observed kinetic energy imparted to the products of the decay corresponds to the decrease in mass. Thus, E is the total relativistic energy of the particle, and m c 2 is its rest energy.

Questions & Answers

plot a graph of MP against tan ( Angle/2) and determine the slope of the graph and find the error in it.
Ime Reply
expression for photon as wave
BARISUA Reply
Are beta particle and eletron are same?
Amalesh Reply
yes
mari
how can you confirm?
Amalesh
sry
Saiaung
If they are same then why they named differently?
Amalesh
because beta particles give the information that the electron is ejected from the nucleus with very high energy
Absar
what is meant by Z in nuclear physic
Shubhu Reply
atomic n.o
Gyanendra
no of atoms present in nucleus
Sanjana
Note on spherical mirrors
Shamanth Reply
what is Draic equation? with explanation
M.D Reply
what is CHEMISTRY
trpathy Reply
it's a subject
Akhter
it's a branch in science which deals with the properties,uses and composition of matter
Eniabire
what is a Higgs Boson please?
FRANKLINE Reply
god particles is know as higgs boson, when two proton are reacted than a particles came out which is used to make a bond between than materials
M.D
bro little abit getting confuse if i am wrong than please clarify me
M.D
the law of refraction of direct current lines at the boundary between two conducting media of
BATTULA Reply
what is the black body of an ideal radiator
Areej Reply
uncertainty principles is applicable to
Areej
fermions
FRANKLINE
what is the cause of the expanding universe?
FRANKLINE
microscopic particles or gases
Areej
Astronomers theorize that the faster expansion rate is due to a mysterious, dark force that is pulling galaxies apart. One explanation for dark energy is that it is a property of space.
Areej
Thanks for your contribution Areej.
FRANKLINE
no problem
Areej
what is photoelectric equation
HIMANSHU Reply
How does fringe intensity depend upon slit width in single slit diffraction?
Abhishek Reply
intensity seems to be directly proportional radius of slit
Mathieu
what are the applications of Bernoulli's equation
Shaukat
VOLTE
Md Reply
what is Draic equation
M.D
about nuclear angular momentum
rahul Reply
what is spin
MUKESH Reply
Practice Key Terms 4

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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