5.9 Relativistic energy (Page 7/16)

University physics volume 3 Page 7 / 16

Key equations

Time dilation	$Δ t = \frac{Δ τ}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} = γ τ$
Lorentz factor	$γ = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$
Length contraction	$L = L_{0} \sqrt{1 - \frac{v^{2}}{c^{2}}} = \frac{L_{0}}{γ}$
Galilean transformation	$x = x' + v t, y = y', z = z', t = t'$
Lorentz transformation	$t = \frac{t' + v x' / c^{2}}{\sqrt{1 - v^{2} / c^{2}}}$
	$x = \frac{x' + v t'}{\sqrt{1 - v^{2} / c^{2}}}$
	$y = y'$
	$z = z'$
Inverse Lorentz transformation	$t' = \frac{t - v x / c^{2}}{\sqrt{1 - v^{2} / c^{2}}}$
	$x' = \frac{x - v t}{\sqrt{1 - v^{2} / c^{2}}}$
	$y' = y$
	$z' = z$
Space-time invariants	${(Δ s)}^{2} = {(Δ x)}^{2} + {(Δ y)}^{2} + {(Δ z)}^{2} - c^{2} {(Δ t)}^{2}$
	${(Δ τ)}^{2} = - {(Δ s)}^{2} / c^{2} = {(Δ t)}^{2} - \frac{[{(Δ x)}^{2} + {(Δ y)}^{2} + {(Δ z)}^{2}]}{c^{2}}$
Relativistic velocity addition	$u_{x} = (\frac{u_{x}^{'} + v}{1 + v u_{x}^{'} / c^{2}}), u_{y} = (\frac{u_{y}^{'} / γ}{1 + v u_{x}^{'} / c^{2}}), u_{z} = (\frac{u_{z}^{'} / γ}{1 + v u_{x}^{'} / c^{2}})$
Relativistic Doppler effect for wavelength	$λ_{obs} = λ_{s} \sqrt{\frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}}$
Relativistic Doppler effect for frequency	$f_{obs} = f_{s} \sqrt{\frac{1 - \frac{v}{c}}{1 + \frac{v}{c}}}$
Relativistic momentum	$\vec{p} = γ m \vec{u} = \frac{m \vec{u}}{\sqrt{1 - \frac{u^{2}}{c^{}}}}$
Relativistic total energy	$E = γ m c^{2}, where γ = \frac{1}{\sqrt{1 - \frac{u^{2}}{c^{2}}}}$
Relativistic kinetic energy	$K_{rel} = (γ - 1) m c^{2}, where γ = \frac{1}{\sqrt{1 - \frac{u^{2}}{c^{2}}}}$

Conceptual questions

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

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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.

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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.

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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.

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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.

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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.

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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.)

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Problems

What is the rest energy of an electron, given its mass is $9.11 \times 10^{- 31} kg ?$ 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.

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Find the rest energy in joules and MeV of a proton, given its mass is $1.67 \times 10^{- 27} kg .$

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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?

$2.3 \times 10^{- 30} kg;$ to two digits because the difference in rest mass energies is found to two digits

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Questions & Answers

differentiate between demand and supply giving examples

Lambiv Reply

differentiated between demand and supply using examples

Lambiv

what is labour ?

Lambiv

how will I do?

Venny Reply

how is the graph works?I don't fully understand

Rezat Reply

information

Eliyee

devaluation

Eliyee

WARKISA

hi guys good evening to all

Lambiv

multiple choice question

Aster Reply

appreciation

Eliyee

explain perfect market

Lindiwe Reply

In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,

Ezea

What is ceteris paribus?

Shukri Reply

other things being equal

AI-Robot

When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab

Kelo

Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)

Kelo

yes,thank you

Shukri

Can I ask you other question?

Shukri

what is monopoly mean?

Habtamu Reply

What is different between quantity demand and demand?

Shukri Reply

Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded

Ezea

Shukri

how do you save a country economic situation when it's falling apart

Lilia Reply

what is the difference between economic growth and development

Fiker Reply

Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.

Shukri

production function means

Jabir

What do you think is more important to focus on when considering inequality ?

Abdisa Reply

any question about economics?

Awais Reply

sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏

Asui

it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs

Awais

thank you so much 👍 sir

Asui

In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p

Cornelius

Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .

Feyisa Reply

Answer

Feyisa

Jabir

the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema

Gsbwnw Reply

suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product

Abdureman

types of unemployment

Yomi Reply

What is the difference between perfect competition and monopolistic competition?

Mohammed

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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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