# 28.6 Relativistic energy  (Page 7/12)

 Page 7 / 12

A photon decays into an electron-positron pair. What is the kinetic energy of the electron if its speed is $0.992c$ ?

$\begin{array}{lll}{\text{KE}}_{\text{rel}}& =& \left(\gamma -1\right){\mathrm{mc}}^{2}=\left(\frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}-1\right){\mathrm{mc}}^{2}\\ & =& \left(\frac{1}{\sqrt{1-\frac{\left(\text{0.992}c{\right)}^{2}}{{c}^{2}}}}-1\right)\left(\text{9.11}×{\text{10}}^{-\text{31}}\phantom{\rule{0.25em}{0ex}}\text{kg}\right)\left(\text{3.00}×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s}{\right)}^{2}=\text{5.67}×{\text{10}}^{-\text{13}}\phantom{\rule{0.25em}{0ex}}\text{J}\end{array}$

## Section summary

• Relativistic energy is conserved as long as we define it to include the possibility of mass changing to energy.
• Total Energy is defined as: $E={\mathrm{\gamma mc}}^{2}$ , where $\gamma =\frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}$ .
• Rest energy is ${E}_{0}={\mathrm{mc}}^{2}$ , meaning that mass is a form of energy. If energy is stored in an object, its mass increases. Mass can be destroyed to release energy.
• We do not ordinarily notice the increase or decrease in mass of an object because the change in mass is so small for a large increase in energy.
• The relativistic work-energy theorem is ${W}_{\text{net}}=E-{E}_{0}=\gamma {\mathrm{mc}}^{2}-{\mathrm{mc}}^{2}=\left(\gamma -1\right){\mathrm{mc}}^{2}$ .
• Relativistically, ${W}_{\text{net}}={\text{KE}}_{\text{rel}}$ , where ${\text{KE}}_{\text{rel}}$ is the relativistic kinetic energy.
• Relativistic kinetic energy is ${\text{KE}}_{\text{rel}}=\left(\gamma -1\right){\mathrm{mc}}^{2}$ , where $\gamma =\frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}$ . At low velocities, relativistic kinetic energy reduces to classical kinetic energy.
• No object with mass can attain the speed of light because an infinite amount of work and an infinite amount of energy input is required to accelerate a mass to the speed of light.
• The equation ${E}^{2}=\left(\mathrm{pc}{\right)}^{2}+\left({\mathrm{mc}}^{2}{\right)}^{2}$ relates the relativistic total energy $E$ and the relativistic momentum $p$ . At extremely high velocities, the rest energy ${\mathrm{mc}}^{2}$ becomes negligible, and $E=\mathrm{pc}$ .

## 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.

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.

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.

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&Exercises

What is the rest energy of an electron, given its mass is $9\text{.}\text{11}×{\text{10}}^{-\text{31}}\phantom{\rule{0.25em}{0ex}}\text{kg}$ ? Give your answer in joules and MeV.

$8.20×{\text{10}}^{-\text{14}}\phantom{\rule{0.25em}{0ex}}\text{J}$

0.512 MeV

Find the rest energy in joules and MeV of a proton, given its mass is $1\text{.}\text{67}×{\text{10}}^{-\text{27}}\phantom{\rule{0.25em}{0ex}}\text{kg}$ .

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 masses in kilograms?

$2\text{.}3×{\text{10}}^{-\text{30}}\phantom{\rule{0.25em}{0ex}}\text{kg}$

what is the formula of displacement
d=vt displacement= velocity×time
bilal
can some one tell me how v=RW is dimensionally correct?
ms-1 = m X Hz
babar
What is displacement
shortest distance b/w two points
bilal
distance+direction
A.d
explain distanace+direction
bilal
the change of postion from one point to another with direction
A.d
if we change thrle direction then displacement is destroy?
bilal
change the direction then?
bilal
what do u mean by i didnt understand bro
A.d
displacement is one dimension...?
bilal
displacement is the total length an object cover from initial to the final with respect to direction as Well as time.
mohammed
thanks
bilal
displacement is the ratio of speed with respect to particular time
Bhautik
shortest distance travel from initial point to final point
ankit
is straight shortest line that connect initial pt with final pt.
Zeleke
what are the differences between vector and scalar quantity
vector is assigned to those physical quantity that has both direction and magnitude! example velocity ,scalar just has magnitude example Mass of an object. hope it helps
Mudang
velocity is produce in fan...?
how many electrons are there in 5 microcouloumb
can a given total amount of mechanical energy be totally converted into heat energy..if so give example
human running
Emmanuel
what is the fumula for calculating specific heat capacity, fusion,fission and vaporization?
Q=cm(∆t)
Emmanuel
Q=cm∆T
what is difference b/w vaporization and evaporation
evaporation is the process of extracting moisture while vaporization is process of becoming a vapor or gas
Emmanuel
From a molecular standpoint they are both cooling processes. Also, you may want to explore states of matter😊 #myTwoCents ~Shi~
Shii
cooling is a similarlity in both process I am confused in difference
1- Evaporation is a process where a liquid change to gas without reaching its boiling point. 2- Vaporization is a process where a liquid change to gas after reaching its boiling point. 3- Sublimation is a process where a solid changes into vapour without passing through a liquid state
Victor
I see. Evaporation is a type of vaporization, that occurs on the surface of a liquid as it changes into the gaseous phase before reaching its boiling point. hope that aids
Shii
vaporisation is cooling process while vaporization is heating process
Emmanuel
I mean to write evaporation is an heating process while vaporization is cooling process
Emmanuel
Yea here are two applications. 1- your wet washed clothes dry under the sun, the water EVAPORATES 2- when u are cooking, it reaches a point where u need to add more water because the water you added previously is getting dried. this is VAPORIZATION. Am not sure which is a cooling or heating process
Victor
vaporization occur only when the evaporation get to level where the above cloud is been (saturated) so cooling take place and started to change to liquid (eg rain fall)
Emmanuel
They are both properties of the same process so they're both cooling
Shii
what about sublimation? cooling or heating process?
Victor
exact
evaporation is the increase in kinetic energy of the liquid which can be gone by adding heat
Emmanuel
so its an heating process
Emmanuel
sublimation is when a solid change to gas
Emmanuel
evaporation is very definitely a cooling process. respectfully@Emmanuel when liquid turns to gas it requires more energy from its surroundings, this energy is in the form of heat, and when heat energy leaves the evaporating liquid it leaves it cooler. Thus, cooling process.
Shii
.
Shii
evaporation is very definitely a cooling process. respectfully@Emmanuel
Shii
kk
Emmanuel
You're right @Shi. I get your point
Victor
eascape velocity on the surface of Earth is 11.2 kms-1 the escape velocity on the surface of another planet of same mass as that of Earth but of 1/4 times of radius of Earth is a5.6kms-1 b11.2 kms-1 c22.4kms-1 d5.6ms-1
Emm.. is that a question? or..
Victor
it is McQ
a)5.6km/s
Alvis
c= Q/cm◇T
A.d
units...
Shii
vital
Shii
the time period of the artificial satellite is given by ?
raza
Why is there no 2nd harmonic in the classical electron orbit?
how to reform magnet after been demagneted
A petrol engine has a output of 20 kilowatts and uses 4.5 kg of fuel for each hour of running. The energy given out when 1 kg of petrol is burnt is 4.8 × 10 to the power of 7 Joules. a) What is the energy output of the engine every hour? b) What is the energy input of the engine every hour?
Issac Newton devised a genius way to calculate changing quantities...
Shii
what is the error during taking work done of a body..
what kind of error do you think? and work is held by which force?
Daniela
I am now in this group
smart
theory,laws,principles and what-a-view are not defined. why? you
A simple pendulum is used in a physics laboratory experiment to obtain an experimental value for the gravitational acceleration, g . A student measures the length of the pendulum to be 0.510 meters, displaces it 10 o from the equilibrium position, and releases it. Using a s
so what question are you passing across... sir?
Olalekan
Two masses of 2 kg and 4 kg are held with a compressed spring between them. If the masses are released, the spring will push them away from each other. If the smaller mass moves off with a velocity of 6m/s, what is the stored energy in the spring when it is compressed?
54 joule
babar
how?
rakesh
Reduce that two body problem into one body problem. Apply potential and k. E formula to get total energy of the system
rakesh
i dont think dere is any potential energy... by d virtue of no height present
Olalekan
there is compressed energy,dats only potential energy na?
rakesh
yes.. but... how will u approach that question without The Height in the question?
Olalekan
Can you explain how you get 54J?
Emmanuel
Because mine is 36J
Emmanuel
got 36J too
Douglas
OK the answer is 54J Babar is correct
Emmanuel
Conservation of Momentum
Emmanuel
woow i see.. can you give the formula for this
joshua