Homework IV-Part 1 Show all work. Draw diagrams and explain answers where appropriate.

1) If the Michelson-Morley experiment is analyzed as if the ether exists, then the round trip time (beam splitter to mirror and back) along the path parallel to the velocity of the trip time(beam splitter to mirror and back) along the parh parallel to the velocity of the lab relative to the ether is given by Δt_par = 2L/c(1-v²/c²)^-1(arm 2 in text). The same quantity for the path perpendicular to v is given by Δt_perp = 2L/c(1-v²/c²)^-1/2 (arm 1 in text). v is the speed of the lab relative to the ether, and L is the distance from the beam splitter to each mirror.

a) Imagine that the experiment was performed with the lasers you used in lab, which emit light with a wavelength of 6.328 x 10^-7 m¹, using a very large interferometer, where L = 3.0 m. Also assume, for calculation purposes, that the speed of the lab relative to the ether is very large. That is, let v = 0.2 c. Determine the total number of cycles that the laser light executes as it passes from the beam splitter to the mirror and back along the "parallel path". Repeat for "perpendicular path".

b) Assume that your results from part a were actually observed. How do they indicate the lab's motion through the ether? How do your predicted results differ from the actual experimental results obtained by Michelson and Morley?

c) The light travels a distance 2L along each path (up and back). According to the formulas given for the elapsed times above is it travelling at the same speed along each path? Explain/Show how you know. What does this have to do with the assumed existence of the ether? Explain how you know.

2) Neutrinos are subatomic particles that have either very little mass or are massless. They are also highly noninteractive and can pass through large amounts of dense matter without being deflected (scattered). It is not unusual, therefore, for a neutrino to pass through the entire Earth from north to south pole with constant velocity. Imagine a neutrino that enters the north pole travelling at 0.998c (relative to Earth) and emerges from the south pole.

a) To an Earth based observer, how long does this take? Is the Earth bound observer measuring a proper time? Explain how you know. Can the proper time be measured in the neutrino rest frame2? Explain how you know. How much time elapses between the two events in the neutrino rest frame?

b) How large is the Earth's diameter as measured from the neutrino rest frame? Is this a proper length? Why or why not?

3) An unstable subatomic particle is observed to have an average lifetime of 4 x 10^-6 s under laboratory conditions where the speed of the particles (many of the same type are observed) is fairly small: v<

a) The scientists on the mountaintop estimate the lifetime of the particle by assuming it can go no faster than c. What value do they calculate? Are they measuring/estimating a "proper time"? Why or why not? (The term "proper" has nothing to do with accuracy.) What about the measurement in the laboratory? Was that a proper time? Why or why not?

b) It's the same particle (this can be determined by other means such as the use of magnetic fields). Why are the two measurements of the lifetime different? i.e. How does relativity explain this?

c) The ratio of the two time measurements can be used to estimate the factor γ without determining the speed. Show the algebra and find the average value of γ for these particles. If you were at rest with an average particle the distance to the mountaintop wouldn't be 6000 m, it would be ...?

4) A space mission is planned to a distant solar system 100 light-years away (that's a distance). The plan is for the rocket carrying a group of astronauts to travel at 0.98c for essentially the entire trip. If this is their speed relative to Earth, so goes relativity theory, the travel time in the rest frame of the astronauts will be much less than that measured from Earth. Since the astronauts will all be in their 30's when they leave the trip will be possible within the life span of the astronauts.

a) If immediately upon arrival the astronauts send the radio message "we're here" back to Earth, how long (in years) after the mission launch will the signal arrive at Earth? Explain the reasoning behind your calculation.

b) How old are the astronauts when they send the signal? Explain the reasoning behind your calculation.