# 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 $0.992c$ ?

## Answer

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

## 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?

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

#### Questions & Answers

Explain why magnetic damping might not be effective on an object made of several thin conducting layers separated by insulation? can someone please explain this i need it for my final exam
anas Reply
Hi
saeid
hi
Yimam
What is thê principle behind movement of thê taps control
Oluwakayode Reply
what is atomic mass
thomas Reply
this is the mass of an atom of an element in ratio with the mass of carbon-atom
Chukwuka
show me how to get the accuracies of the values of the resistors for the two circuits i.e for series and parallel sides
Jesuovie Reply
Explain why it is difficult to have an ideal machine in real life situations.
Isaac Reply
tell me
Promise
what's the s . i unit for couple?
Promise
its s.i unit is Nm
Covenant
Force×perpendicular distance N×m=Nm
Oluwakayode
İt iş diffucult to have idêal machine because of FRİCTİON definitely reduce thê efficiency
Oluwakayode
if the classica theory of specific heat is valid,what would be the thermal energy of one kmol of copper at the debye temperature (for copper is 340k)
Zaharadeen Reply
can i get all formulas of physics
BPH Reply
yes
haider
what affects fluid
Doreen Reply
pressure
Oluwakayode
Dimension for force MLT-2
Promise Reply
what is the dimensions of Force?
Osueke Reply
how do you calculate the 5% uncertainty of 4cm?
melia Reply
4cm/100×5= 0.2cm
haider
how do you calculate the 5% absolute uncertainty of a 200g mass?
melia Reply
= 200g±(5%)10g
haider
use the 10g as the uncertainty?
melia
which topic u discussing about?
haider
topic of question?
haider
the relationship between the applied force and the deflection
melia
sorry wrong question i meant the 5% uncertainty of 4cm?
melia
its 0.2 cm or 2mm
haider
thank you
melia
Hello group...
Chioma
hi
haider
well hello there
sean
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Noks
hii
Chibueze
10g
Olokuntoye
0.2m
Olokuntoye
hi guys
thomas
the meaning of phrase in physics
Chovwe Reply
is the meaning of phrase in physics
Chovwe
write an expression for a plane progressive wave moving from left to right along x axis and having amplitude 0.02m, frequency of 650Hz and speed if 680ms-¹
Gabriel Reply
how does a model differ from a theory
Friday Reply
To use the vocabulary of model theory and meta-logic, a theory is a set of sentences which can be derived from a formal model using some rule of inference (usually just modus ponens). So, for example, Number Theory is the set of sentences true about numbers. But the model is a structure together wit
Jesilda
with an iterpretation.
Jesilda

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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