The proper time interval

Problem:

A possible clock is shown in the figure below.  It consists of a flashtube F and a photocell P shielded so that each views only the mirror M, located a distance d away, and mounted rigidly with respect to the flashtube-photocell assembly.  The electronic innards of the box are such that, when the photocell responds to a light flash from the mirror, the flashtube is triggered with a negligible delay and emits a short flash towards the mirror.  The clock thus "ticks" once every (2d/c) seconds when at rest.

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(a) Suppose that the clock moves with a uniform velocity v, perpendicular to the line from PF to M, relative to an observer.  Using the second postulate of relativity, show by explicit geometrical or algebraic construction that the observer sees the relativistic time dilatation as the clock moves by.
(b) Suppose that the clock moves with a velocity v parallel to the line from PF to M.  Verify that here, too, the clock is observed to tick more slowly, by the same time dilatation factor.

Solution:

Problem:

Consider a light bulb inside a train moving with velocity v with respect to the train station.  The bulb is switched on, and its light hits the floor of the wagon right underneath the bulb.  The bulb-floor distance is h.  Using only the postulate of relativity about the speed of light, calculate the time it takes for the light to hit the floor from the perspective of an observer at rest inside the train and from the perspective of an observer at rest at the station and compare those times.

Solution:

Problem:

The half-life of a π+ meson at rest is 2.5*10-8 s.   A beam of π+ mesons is generated at a point 15 m from a detector.  Only ½ of the π+ mesons live to reach the detector.  What is the speed of the π+ mesons?

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Problem:

How far will a muon (whose lifetime is 2.2 µsec in its rest frame) move before it decays in the lab if it moving at 0.999c?

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Problem:

An atomic clock is taken to the North Pole, while another stays at the Equator.  How far will they be out of synchronization after a year has elapsed?

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Problem:

The premise of the Planet of the Apes movies and book is that hibernating astronauts travel far into Earth's future, to a time when human civilization has been replaced by an ape civilization. Considering only special relativity, determine how far into Earth's future the astronauts would travel if they slept for 120 years while traveling relative to the Earth with a speed of v = 0.9990c, first outward from Earth and then back again.

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Problem:

You wish to make a round trip from Earth in a spaceship, traveling to a distant planet at constant speed in a straight line for exactly 6 months (as you measure the time interval in your rest frame) spend 1 year at on the planet, and return at the same constant speed.  Assume the planet is at rest with respect to Earth.
You wish further, on your return, to find Earth as it will be exactly 1001 years in the future as measured on Earth.  At what speed v must you travel?  (Give your answer as a fraction of c.)

Solution:

Problem:

The radius of the galaxy is 3*1020 m, measured in its own rest frame.
(a)  If the time it takes a spaceship to cross the entire galaxy is 300 years measured in the spaceship's rest frame, what is the relative speed of spaceship and galaxy?
(b)  How much time elapses on Earth during this trip?

Problem:

A space ship has a proper length of 100 m.  It travels close to the Earth's surface with a constant speed of 0.8 c.  Earth observers decide to measure the length of the ship by erecting two towers that coincide with the ends of the ship simultaneously (in the Earth's frame) as it passes by.
(a)  How far apart do the observers on Earth build the towers?
(b)  How long do the observers on Earth say it takes for the nose of the ship to travel from tower A to tower B?
(c)  How long, according to the measurements in the spaceship frame, does it take for the nose of the ship to travel from tower A to tower B?
(d)  As measured by passengers in the spaceship, how far apart are the two towers?
(e)  In the spaceship frame, how long does it take a beam of light to travel from the front to the rear of the spaceship?
(f)  How much time, according to the observers on Earth, is required for a beam of light to travel from the front to the rear of the moving spaceship?
(Give numerical answers!)

Problem:

A space-time diagram is a plot of position versus time.  It is always drawn in an inertial frame.  Consider the space-time diagram shown below.

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(a) Assume a particle moves along the path OPB from O to B.   Describe the motion in the inertial frame in which the space-time diagram is drawn.  Find the proper time interval (the time interval in the rest frame of the particle) between events O and B.
(b) Assume a particle moves along the path ORB from O to B.   Describe the motion in the inertial frame in which the space-time diagram is drawn.  Find the proper time interval (the time interval in the rest frame of the particle) between events O and B.
(c) Comment on the implication for space travel that you can deduce from your analysis of this simple example.

Solution:

Problem:

Twins Alice and Bob, who are 19 years old, leave the earth and travel to a distant planet 12 light-years away.  Assume that the planet and earth are at rest with respect to each other.  The twins depart at the same time on different spaceships.  Alice travels at a speed of 0.5c, and Bob travels at 0.9c. 
(a)  What is the difference between their ages when they meet again on earth at the earliest possible time, and which twin is older?
(b)  If instead of traveling at a constant speed of 0.5c, Alice had covered the total distance in the same time, but on her trip to and from the planet had accelerated and travelled 50% of the time with speed v and 50% of the time with speed of 3v, as measured on earth, how old would she be when she meets Bob again.

Solution:

Problem:

Assume a rocket ship leaves the earth in the year 2020.  One of a set of twins born in 2000 remains on earth, the other rides in the rocket.  The rocket ship is so constructed that it has an acceleration g in its own rest frame to make the occupants feel at home.  It accelerates in a straight line path for 5 years as measured by its own clock, decelerates at the same rate for 5 more years years, turns around, accelerates for 5 years, decelerates for 5 years, and lands on earth.  The twin in the rocket is 40 years old.
(a)  What year is it on earth?
(b)  How far away from earth did the rocket ship travel?

Solution:

Problem:

A monochromatic particle beam consist of particles whose total energy is 100 times their rest energy.  The rest lifetime of the particles is 0.10 ns.  In the laboratory, the distance between the point where the particles are generated and the detector is 6.0 m.  
(a)  What is the speed of the particles?
(b)  What fraction of the generated particles reaches the detector?

Solution: