Chapter 35

Michelson and Morley used an interferometer to show that the speed of light is constant, regardless of Earth's motion through any perceived luminiferous aether. An analogy can be understood from the different times it takes for a rowboat to travel two different round-trip paths in a river that flows at a constant velocity \((u)\) downstream. Let one path be for a distance \(D\) directly across the river, then back again; and let the other path be the same distance \(D\) directly upstream, then back again. Assume that the rowboat travels at constant speed, \(v\) (with respect to the water), for both trips. Neglect the time it takes for the rowboat to turn around. Find the ratio of the cross-stream time divided by the upstream-downstream time, as a function of the given constants.

What is the value of \(\gamma\) for a particle moving at a speed of \(0.8 c ?\)

An astronaut on a spaceship traveling at a speed of \(0.50 c\) is holding a meter stick parallel to the direction of motion. a) What is the length of the meter stick as measured by another astronaut on the spaceship? b) If an observer on Earth could observe the meter stick, what would be the length of the meter stick as measured by that observer?

A spacecraft travels along a straight line from Earth to the Moon, a distance of \(3.84 \cdot 10^{8} \mathrm{~m}\). Its speed measured on Earth is \(0.50 c\). a) How long does the trip take, according to a clock on Earth? b) How long does the trip take, according to a clock on the spacecraft? c) Determine the distance between Earth and the Moon if it were measured by a person on the spacecraft.

If a muon is moving at \(90.0 \%\) of the speed of light, how does its measured lifetime compare to when it is in the rest frame of a laboratory, where its lifetime is \(2.2 \cdot 10^{-6}\) s?

In Jules Verne's classic Around the World in Eighty Days, Phileas Fogg travels around the world in, according to his calculation, 81 days. Due to crossing the International Date Line he actually made it only 80 days. How fast would he have to go in order to have time dilation make 80 days to seem like \(81 ?\) (Of course, at this speed, it would take a lot less than even 1 day to get around the world \(\ldots . .)\)

Suppose NASA discovers a planet just like Earth orbiting a star just like the Sun. This planet is 35 light-years away from our Solar System. NASA quickly plans to send astronauts to this planet, but with the condition that the astronauts would not age more than 25 years during this journey. a) At what speed must the spaceship travel, in Earth's reference frame, so that the astronauts age 25 years during this journey? b) According to the astronauts, what will be the distance of their trip?

A wedge-shaped spaceship has a width of \(20.0 \mathrm{~m}\) a length of \(50.0 \mathrm{~m},\) and is shaped like an isosceles triangle. What is the angle between the base of the ship and the side of the ship as measured by a stationary observer if the ship is traveling by at a speed of \(0.400 c\) ? Plot this angle as a function of the speed of the ship.

In your physics class you have just learned about the relativistic frequency shift, and you decide to amaze your friends at a party. You tell them that once you drove through a stop light and that when you were pulled over you did not get ticketed because you explained to the police officer that the relativistic Doppler shift made the red light of wavelength \(650 \mathrm{nm}\) appear green to you, with a wavelength of \(520 \mathrm{nm}\). If your story had been true, how fast would you have been traveling?

A meteor made of pure kryptonite (Yes, we know: There really isn't such a thing as kryptonite ...) is moving toward Earth. If the meteor eventually hits Earth, the impact will cause severe damage, threatening life as we know it. If a laser hits the meteor with wavelength \(560 \mathrm{nm}\), the entire meteor will blow up. The only laser powerful enough on Earth has a \(532-\mathrm{nm}\) wavelength. Scientists decide to launch the laser in a spacecraft and use special relativity to get the right wavelength. The meteor is moving very slowly, so there is no correction for relative velocities. At what speed does the spaceship need to move so the laser has the right wavelength, and should it travel toward or away from the meteor?