# 5.9 Relativistic energy  (Page 7/16)

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## Key equations

 Time dilation $\text{Δ}t=\frac{\text{Δ}\tau }{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}=\gamma \tau$ Lorentz factor $\gamma =\frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}$ Length contraction $L={L}_{0}\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}=\frac{{L}_{0}}{\gamma }$ Galilean transformation $x=x\prime +vt,\phantom{\rule{0.5em}{0ex}}y=y\prime ,\phantom{\rule{0.5em}{0ex}}z=z\prime ,\phantom{\rule{0.5em}{0ex}}t=t\prime$ Lorentz transformation $t=\frac{t\prime +vx\prime \text{/}{c}^{2}}{\sqrt{1-{v}^{2}\text{/}{c}^{2}}}$ $x=\frac{x\prime +vt\prime }{\sqrt{1-{v}^{2}\text{/}{c}^{2}}}$ $y=y\prime$ $z=z\prime$ Inverse Lorentz transformation $t\prime =\frac{t-vx\text{/}{c}^{2}}{\sqrt{1-{v}^{2}\text{/}{c}^{2}}}$ $x\prime =\frac{x-vt}{\sqrt{1-{v}^{2}\text{/}{c}^{2}}}$ $y\prime =y$ $z\prime =z$ Space-time invariants ${\left(\text{Δ}s\right)}^{2}={\left(\text{Δ}x\right)}^{2}+{\left(\text{Δ}y\right)}^{2}+{\left(\text{Δ}z\right)}^{2}-{c}^{2}{\left(\text{Δ}t\right)}^{2}$ ${\left(\text{Δ}\tau \right)}^{2}=-{\left(\text{Δ}s\right)}^{2}\text{/}{c}^{2}={\left(\text{Δ}t\right)}^{2}-\frac{\left[{\left(\text{Δ}x\right)}^{2}+{\left(\text{Δ}y\right)}^{2}+{\left(\text{Δ}z\right)}^{2}\right]}{{c}^{2}}$ Relativistic velocity addition ${u}_{x}=\left(\frac{{u}_{x}^{\prime }+v}{1+v{u}_{x}^{\prime }\text{/}{c}^{2}}\right),\phantom{\rule{0.5em}{0ex}}{u}_{y}=\left(\frac{{u}_{y}^{\prime }\text{/}\gamma }{1+v{u}_{x}^{\prime }\text{/}{c}^{2}}\right),\phantom{\rule{0.5em}{0ex}}{u}_{z}=\left(\frac{{u}_{z}^{\prime }\text{/}\gamma }{1+v{u}_{x}^{\prime }\text{/}{c}^{2}}\right)$ Relativistic Doppler effect for wavelength ${\lambda }_{\text{obs}}={\lambda }_{s}\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}$ Relativistic Doppler effect for frequency ${f}_{\text{obs}}={f}_{s}\sqrt{\frac{1-\frac{v}{c}}{1+\frac{v}{c}}}$ Relativistic momentum $\stackrel{\to }{p}=\gamma m\stackrel{\to }{u}=\frac{m\stackrel{\to }{u}}{\sqrt{1-\frac{{u}^{2}}{{c}^{}}}}$ Relativistic total energy $E=\gamma m{c}^{2},\phantom{\rule{0.2em}{0ex}}\text{where}\phantom{\rule{0.2em}{0ex}}\gamma =\frac{1}{\sqrt{1-\frac{{u}^{2}}{{c}^{2}}}}$ Relativistic kinetic energy ${K}_{\text{rel}}=\left(\gamma -1\right)m{c}^{2},\phantom{\rule{0.2em}{0ex}}\text{where}\phantom{\rule{0.2em}{0ex}}\gamma =\frac{1}{\sqrt{1-\frac{{u}^{2}}{{c}^{2}}}}$

## Conceptual questions

How are the classical laws of conservation of energy and conservation of mass modified by modern relativity?

What happens to the mass of water in a pot when it cools, assuming no molecules escape or are added? Is this observable in practice? Explain.

Because it loses thermal energy, which is the kinetic energy of the random motion of its constituent particles, its mass decreases by an extremely small amount, as described by energy-mass equivalence.

Consider a thought experiment. You place an expanded balloon of air on weighing scales outside in the early morning. The balloon stays on the scales and you are able to measure changes in its mass. Does the mass of the balloon change as the day progresses? Discuss the difficulties in carrying out this experiment.

The mass of the fuel in a nuclear reactor decreases by an observable amount as it puts out energy. Is the same true for the coal and oxygen combined in a conventional power plant? If so, is this observable in practice for the coal and oxygen? Explain.

Yes, in principle there would be a similar effect on mass for any decrease in energy, but the change would be so small for the energy changes in a chemical reaction that it would be undetectable in practice.

We know that the velocity of an object with mass has an upper limit of c . Is there an upper limit on its momentum? Its energy? Explain.

Given the fact that light travels at c , can it have mass? Explain.

Not according to special relativity. Nothing with mass can attain the speed of light.

If you use an Earth-based telescope to project a laser beam onto the moon, you can move the spot across the moon’s surface at a velocity greater than the speed of light. Does this violate modern relativity? (Note that light is being sent from the Earth to the moon, not across the surface of the moon.)

## Problems

What is the rest energy of an electron, given its mass is $9.11\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-31}\phantom{\rule{0.2em}{0ex}}\text{kg}?$ Give your answer in joules and MeV.

0.512 MeV according to the number of significant figures stated. The exact value is closer to 0.511 MeV.

Find the rest energy in joules and MeV of a proton, given its mass is $1.67\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-27}\phantom{\rule{0.2em}{0ex}}\text{kg}.$

If the rest energies of a proton and a neutron (the two constituents of nuclei) are 938.3 and 939.6 MeV, respectively, what is the difference in their mass in kilograms?

$2.3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-30}\phantom{\rule{0.2em}{0ex}}\text{kg};$ to two digits because the difference in rest mass energies is found to two digits

plot a graph of MP against tan ( Angle/2) and determine the slope of the graph and find the error in it.
expression for photon as wave
Are beta particle and eletron are same?
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
atomic n.o
Gyanendra
no of atoms present in nucleus
Sanjana
Note on spherical mirrors
what is Draic equation? with explanation
what is CHEMISTRY
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?
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
what is the black body of an ideal radiator
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
FRANKLINE
no problem
Areej
what is photoelectric equation
How does fringe intensity depend upon slit width in single slit diffraction?
intensity seems to be directly proportional radius of slit
Mathieu
what are the applications of Bernoulli's equation
Shaukat
VOLTE
what is Draic equation
M.D
what is spin