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Energy, Gen’rl Rel., & Space-time diagrams, Class 4: (ThT Q) Does Kinetic Energy still have a meaning in relativity? What is rest energy? What does E= mc2 mean? How does the equation come into being?
Did you complete at least 70% of Chapter 1: 8-10 & Appendix 2 and view the assigned MU 44: 9’15” to 14’ 30”? • Yes • No
Energy (moving particle) E = γ mc2 Rest Energy: E0 = mc2 Kinetic Energy: K = mc2 (γ -1) Rest Energy Electron: mc2 = 0.511 MeV Proton: mc2 = 938 MeV
Consider an electron and a proton, each with a kinetic energy of 1 MeV. Which particle is moving at relativistic speeds? (pp) • electron • proton Let’s do the work
Electron K = 1MeV • Classical K = ½ mv2 v≈ 2c • Relativity K = mc2 (γ -1)γ= k/mc2 +1 = 1MeV/[0.511MeV+1] = 2.95 • γ = 1/√[1-v2/c2] • v = c√[1-1/γ2] = 0.94c
Proton K = 1MeV • Classical: K = ½ mv2v =0.046c • Relativity: • γ = 1MeV/938MeV +1 = 1.001066 • V = c√[1-1/γ2] = 0.046c What do you know!!
Neutron: as cosmic ray • K = 50 J ≈ 3 x 1020 eV • γ = K/mc2 + 1 = 6 x 1014 • v = c√[1-1/γ2]; γ-2 is as close to 0 as you are likely to get, so do binomial expansion. • ≈ c[1-½ (1/γ2)]; keeping only 1st term • 0.999999999999999999999999999985c
What is the lifetime for the speedy neutron in lab frame? • Basic facts: neutrons which are not in a nucleus are unstable decaying to a proton, an electron and an antineutrino. The mean half life is 886 s. • γ*886s = 5 x 1017s = 1.5 x 1010 years, life of the Universe.
Suppose that the distance between stars is about 4-light years. • What is distance between stars in the frame of this relativistic particle? • L = 4 x1016m/γ = 4 x1016m/6x1014 • = 67 m. • Can you imagine stars that close?
What happens to the Energy when the particle hits an atom in the earth’s atmosphere? • What does this have to do with E = mc2?
Converting Mass to Energy • Fusion and fission.
1. (a) What is the total energy of a particle with rest mass of one gram moving at half the speed of light? (b) A 20-megaton hydrogen bomb explodes and releases the same energy as 20 million tons of TNT. Given the equivalence that 1 ton of TNT equals 4.18 × 109J, how much mass was used up in the explosion?
2. (2 pts) (a) A particle with a rest energy of 2400 MeV has an energy of 15 GeV. Find the time (in Earth's frame of reference) necessary for this particle to travel from Earth to a star four light-years distant. (See the front of your text book for the definition of 1 eV.) (b) How much work (in MeV) must be done to increase the speed of an electron from 1.2x108m/s to 2.4x108m/s?
3. (2 points) Twins Bill and Abby separate. Abby travels at a uniform speed (except for a brief turn-around period) to a star 4.2 light-years away, and upon returning, finds she is 5.2 years younger than Bill. (a) How fast was Abby traveling? (b) By how much has Bill aged during the period Abby was traveling?
4. (2 pts) Rocket A moves with a speed of 0.75c in a northerly direction relative to the origin. Rocket B moves to the west relative to the origin with speed 0.4c. How fast and in what direction does the pilot of Rocket A observe rocket B to be traveling?
5. (2 pts) An observer detects two explosions that occur at the same time, one near her and another 100 km away. Another observer finds that the two explosions occur 160 km apart. What time interval separates the two explosions according to the second observer?
6. (2 pts) Consider the births and deaths of three people as seen from two different inertial frames S and S0. S0 moves along the x-axis at a speed of 0.98c with respect to S. At t = t0 = 0, observers in S record the simultaneous births of three people: Jack, born at the origin; Astrid, born at x = +10 ly; and Jill, born at the origin. Both Jack and Astrid are always at rest with respect to S throughout their 70-year lifespan. Jill is born and remains on a spaceship, which is at rest with respect to S0. With respect to S0, Jill also lives 70 years. According to observers in S, how long does Jill live?
General Relativity (includes gravity) • Equivalence Principle:Object in free fall in equivalent to an inertial frame of reference without gravity. • Acceleration is the same as gravity.
Consider a box in free-fall. A laser in the box shoots a beam from one side of the box to the other. In the ref. frame of the box, the laser beam will be: (pp) • curved upward • curved downward • not curved
Consider the space shuttle in orbit around the Earth. Is the shuttle in an inertial frame of reference? (pp) • yes • no
Gravity disappears • Space tells matter how to move. • Matter tells space how to curve. • John Wheeler. • Question: • Can the other Forces disappear?
EXTRA CREDIT (2 pts) SIM: Relativistic Movies • Watch the simulations of relativistic effects at http://www.anu.edu.au/physics/Savage/TEE/gallery.html • Write a paragraph about what you observed or learned about relativistic effects while watching these movies. You can select a single movie to discuss or use example from multiple movies to describe what you observed.
Relativity • F = dp/dt • p = γmv = relativistic momentum conserved in collisions • γ = 1/√[1-v2/c2] • v«c γ ≈ 1 p = mv • v c γ ∞ p ∞