Environmental Engineering Reference
In-Depth Information
stating the same result is to say that the total energy is almost entirely made up of
rest mass energy for a non-relativistic system.
Example 7.2.2
For the electron in the TV set of the previous example, estimate its
speed using non-relativistic mechanics.
Solution 7.2.2
Since the electron has a kinetic energy of 20.0 keV we might expect
to be able to use non-relativistic mechanics since 20.0 keV is much smaller than
the electron rest mass energy of 511 keV. Let us see how good an approximation it
actually is. In the non-relativistic limit, energy conservation dictates that
1
2
mv
2
eV
2
eV
m
⇒
v
40
.
0
511
c
=
0
.
280
c.
This agrees with the full relativistic result to an accuracy of
4%
.
One final remark is in order. If you do decide to simplify a problem by working
with the formulae of non-relativistic mechanics and the result is a speed which is
comparable to the speed of light, or an energy which is comparable to a rest mass
energy, then you were wrong to employ the non-relativistic approximation and need
to start over but this time with the correct relativistic expressions. Conversely, if the
speeds you obtain are small compared to the speed of light then the non-relativistic
approximation was good and you can be sure of your results.
Example 7.2.3
In particle physics experiments, physicists routinely accelerate sub-
atomic particles through enormous voltages before making them collide with each
other. The particle kinetic energies are often much larger than their rest mass ener-
gies which means that the situation is extremely relativistic. It also means that very
many new particles can be created out of a single collision.
For example, the Large Hadron Collider at CERN will make head-on collisions
between pairs of protons. Each proton will have an energy of 7000 GeV (
1
GeV
=
1000
MeV).
5
The main goal of the LHC is to convert this kinetic energy into the
mass of new, hitherto undiscovered, particles such as Higgs bosons or supersym-
metric particles.
(a) What is the rest energy in MeV of a proton given that each proton has a mass
of
1
.
67
10
−
27
kg?
(b) What speed are the LHC protons travelling at?
(c) What is the momentum of an LHC proton?
(d) How many new protons can in principle be produced in a single collision?
×
5
You might like to convince yourself that this is the roughly equal to the kinetic energy of a tennis ball
travelling at
∼
5 mm/s.
