Review: Relativistic mechanics. Announcements. Relativistic kinetic energy. Kinetic energy. E tot = γmc 2 = K + mc 2. K = γmc 2 - mc 2 = (γ-1)mc 2

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1 Announceents Reading for Monday: Chapters Review session for the idter: in class on Wed. HW 4 due Wed. Exa 1 in 6 days. It covers Chapters 1 & 2. Roo: G1B30 (next to this classroo). Review: Relativistic echanics Relativistic oentu: Relativistic force: Total energy of a particle with ass : E tot = γc 2 = K + c 2 These definitions fulfill the oentu and energy conservation laws. And for u<<c the definitions for p, F, and K atch the classical definitions. But we found that funny stuff happens to the proper ass. Kinetic energy The work done by a force F to ove a particle fro position 1 to 2 along a path s is defined by: Relativistic kinetic energy The relativistic kinetic energy K of a particle with a rest ass is: K = γc 2 - c 2 = (γ-1)c 2 K 1,2 being the particle's kinetic energy at positions 1 and 2, respectively (true only for frictionless syste!). Using our prior definition for the relativistic force we can now find the relativistic kinetic energy of the particle. Note: This is very different fro the classical K= ½v 2. For slow velocities the relativistic energy equation gives the sae value as the classical equation! Reeber the binoial approxiation for γ: γ 1+ ½v 2 /c 2 K = (γ-1)c 2 (1 + ½v 2 /c 2 1)c 2 = ½ v 2

2 Total energy We rewrite the equation for the relativistic kinetic energy and define the total energy of a particle as: E = γc 2 = K + c 2 QUIZ: Rest energy of a particle E = γc 2 = K + c 2 In the particle's rest frae, its energy is its rest energy, E 0. What is the value of E 0? This definition of the relativistic ass-energy E fulfills our condition of conservation of total energy. (Not proven here, but we shall see several exaples where this proves to be correct.) A: 0 B: c 2 C: c 2 D: (γ-1)c 2 E: ½ c 2 Note: This suggests a connection between ass and energy! Relation between Mass and Energy Equivalence of Mass and Energy v -v v -v Conservation of the total energy requires that the final energy E tot,final is the sae as the energy E tot, before the collision. Therefore: E 1 = γc 2 = K + c 2 E 2 = γc 2 = K + c 2 E tot = γ final Mc2 = 2K initial + 2c2 1 Total energy: E tot = E 1 +E 2 = 2K + 2c 2 E tot,final = Mc 2 2K + 2c 2 = E tot,initial We find that the total ass M of the final syste is larger than the su of the asses of the two parts! M>2. Potential energy inside an object contributes to its ass!!!

3 The vibrating spring a) Has a constant ass b) Has the sallest ass when it is oving fastest (in the iddle of its otion) c) Has the sallest ass when it is oving slowest (at the end of its otion) Exaple: Rest energy of an object with 1kg E 0 = c 2 = (1 kg) ( / s ) 2 = J J = kwh = 2.9 GW 1 year This is a very large aount of energy! (Equivalent to the yearly output of ~3 very large nuclear reactors.) Enough to power all the hoes in Colorado for a year! Way to convert ass to energy

4 Exaple: Deuteriu fusion Isotopes of Hydrogen: Exaple: Deuteriu fusion Isotope ass: Deuteriu: u Heliu 4: u (1 u kg) 1kg of Deuteriu yields ~0.994 kg of Heliu 4. Energy equivalent of 6 gras: E 0 = c 2 = (0.006 kg) ( / s ) 2 = J Enough to power ~20,000 Aerican households for 1 year! Relationship of Energy and oentu Recall: Total Energy: E = γc 2 Moentu: p = γu Therefore: p 2 c 2 = γ 2 2 u 2 c 2 = γ 2 2 c 4 u 2 /c 2 use: Mc 2 = Σ( i c 2 ) E B p 2 c 2 = γ 2 2 c 4 2 c 4 =E 2 This leads us the oentu-energy relation: or: E 2 = (pc) 2 + (c 2 ) 2 E 2 = (pc) 2 + E 0 2

5 Application: Massless particles Fro the oentu-energy relation E 2 = p 2 c c 4 we obtain for ass-less particles (i.e. =0): E = pc, (if =0) p=γu and E=γc 2 p/u = E/c 2 Using E=pc leads to: u=c, (if =0) Massless particles travel at the speed of light!! no atter what their total energy is!! Exaple: Electron-positron annihilation Positrons (e +, aka. antielectron) have exactly the sae ass as electrons (e - ) but the opposite charge: e+ = e- = 511 kev/c 2 (1 ev J) ebam! - e + E 1, p 1 E 2, p 2 At rest, an electron-positron pair has a total energy E = kev. Once they coe close enough to each other, they will annihilate one another and convert into two photons. Conservation What can of you oentu: tell about p those 1 = -p 2 two. photons? Conservation of energy: E 1 +E 2 = 2c 2, E 1 = E 2 = 511 kev Do neutrinos have a ass? Neutrinos are eleentary particles. They coe in three flavors: electron, uon, and tau neutrino (ν e,ν µ, ν τ ). The standard odel of particle physics predicted such particles. The prediction said that they were ass-less. The fusion reaction that takes place in the sun (H + H He) produces such ν e. The standard solar odel predicts the nuber of ν e coing fro the sun. All attepts to easure this nuber on earth revealed only about one third of the nuber predicted by the standard solar odel. Do neutrinos have a ass? (cont.) Bruno Pontecorvo predicted the neutrino oscillation, a quantu echanical phenoenon that allows the neutriono to change fro one flavor to another while traveling fro the sun to the earth. Why would this iply that the neutrinos have a ass? Massless particles travel at the speed of light! i.e. γ, and therefore, the tie sees to be standing still for the neutrino: Δt Earth = γ Δt neutrino( proper ) In the HW: uon or pion experients. The half-live tie of the uons/pions in the lab-frae is increased by the factor γ.

6 Suary SR Classical relativity Galileo transforation Special relativity (consequence of 'c' is the sae in all inertial fraes; reeber Michelson-Morley experient) Tie dilation & Length contraction, events in spacetie Lorentz transforation Spacetie interval (invariant under LT) Relativistic forces, oentu and energy Lot's of applications (and lot's of firecrackers) Everything we have discussed to this point will be part of the first id-ter exa (including reading assignents and hoework.) If you have questions ask as early as possible!!