Answer as completely as you can. No equations are needed, but well-labeled diagrams will be needed to have a complete solution. You may need to go to sources other than your class notes and text to find information to completely answer these questions. (Be sure to cite any sources that you use!)
(a) Describe and explain the set-up of the Michelson-Morley experiment.
(b) Describe a bit of the history of the luminiferous ether, and explain how the interference of light was used to try to prove the existence of the ether.
(c) Discuss the results of the experiment and the conclusion drawn from those results.
(d) What was the impact of the conclusions drawn from this experiment?
Extra problem 1.B
You take a tennis ball and throw it against the wall, and then catch it when it bounces back. Say that you are standing a distance D from the wall, and the positive x-axis points horizontally from the wall toward you. Sketch a classical spacetime diagram (t vs. x) showing the motion of the ball from when you threw it to when you caught it. (Be careful to label relevant information.) Note: We are dropping the “c” from the “ct”-axis so that the (very) non-relativistic speeds in your diagram can have “normal” slopes.
Extra problem 1.C
You take a tennis ball and drop it vertically from a height relatively large H onto a carpeted floor. The ball bounces five times before it effectively stops. Take the positive-y direction to be vertically downwards. Sketch a classical spacetime diagram (t vs. y) showing the motion of the ball from when you dropped it to when it stopped. (Be careful to label relevant information.) Note: We are dropping the “c” from the “ct”-axis so that the (very) non-relativistic speeds in your diagram can have “normal” slopes.
Extra problem 1.D
Do CH 1, problem 10 parts (a) and (b) in the text. Sketch ST diagrams to support your answers. Instead of the text’s part (c), do the following:
(c) If you answered “no” in parts (a) or (b), state which observer detects the light signal first.
Extra problem 1.E
The figure below is taken from the Addition of Velocities PowerPoint presentation with the addition of the point F. (The point was there before, we just didn’t label it!) This diagram represents the relativistic addition of velocities.
(a) Argue that the ratio of CB to CF is just bux’ = u'x/c. That is, argue that the ratio of CB to CF is just the (dimensionless) fraction of the speed of light traveled by the object as observed in the S’ reference frame.
(b) Now imagine that the object in (a) throws another object forward with some speed that is some fraction of c. Argue that the speed of this new object as viewed in S (ie, relative to the ct-x axes above) is still less than c.
(c) Extend the argument in (b) to any number of objects thrown by other objects, all moving faster and faster as seen by S, but the speed of the final object thrown is always less than c.
(d) Use these arguments to deduce that no material object can travel faster than c.
Extra problem 1.F
Using algebra, solve the relativistic inverse velocity transformation equation to obtain the relativistic velocity transformation equation, thereby verifying the argument made in class.
Extra problem 1.G
The figure above in problem 1.E (or equivalently, the figures in the Addition of Velocities PowerPoint presentation) is drawn to scale. Using this diagram, find
(a) the velocity of the S’ frame relative to the “stationary” S frame. (Hint: you’ll need a protractor.)
(b) the velocity of the object relative to the “stationary” S reference frame.
(c) Use the relativistic velocity transformation equation to find the velocity of the object as viewed by the moving S’ reference frame. Does this answer seem reasonable looking at the ST diagram?
Extra problem 1.H
A space platform is the staging area for two relativistic rockets. The first rocket leaves the platform in a given direction with a speed of 0.90c. Later, a second rocket leaves the platform at a speed of 0.98c in the same direction as the first rocket.
(a) What is the relative velocity of the second rocket as seen by the first?
(b) Answer question (a) if the second “rocket” were a beam of light instead of a rocket. (That is, instead of firing a second rocket, we fire a beam of light.)
Extra problem 1.I
A man on the moon sees two spacecraft, A and B, moving directly toward him from opposite directions. The first spacecraft moves with a speed of 0.800c, while the second moves with a speed of 0.900c.
(a) What does a spacewoman on A measure for the speed with which she is approaching the moon?
(b) What does a spacewoman on A measure for the speed with which she is approaching spacecraft B?
(c) What does a spaceman on B measure for the speed with which he is approaching the moon?
(d) What does a spaceman on B measure for the speed with which he is approaching spacecraft A?
Scaled ST Diagrams
Work through the Drawing Spacetime Diagrams presentation that can be accessed by clicking on the link below.
Then draw a scaledspacetime diagram corresponding to β = 0.62. This should be similar to what we’ve been seeing in class, with the x-ct and x’-ct’ axes on the same graph, except that both the x-ct and x’-ct’ axes should extend equally to positive and negative regions. (You will probably want to do this on some good graph paper or on a computer. Make your diagram as large as possible!) Draw tick-marks on your axes with appropriate scaled distances between the tick marks on the unprimed and primed axes, but don’t label numbers on the tick marks to leave your diagram flexible so you can use copies of it to solve the problems below. (So you can change your axes, for example, to run from 0 to 5 m for one problem, and from 0 to 20 m for another. Read the problems below to see how your diagram copies will be used.) Make some copies of your diagram before you start working on the problems below. Each part in the problems below should be answered on a different copy of your ST diagram.)
Extra Problem 1.M Superman!
(a) A 2.7 m long car sits at rest along the positive x-axis with one end at the origin. Superman!passes us and the car flying at 0.62 c in the positive x-direction. How long is the car as measured by Superman! ? Answer this question as measured to scale on your ST diagram, and then do the calculation to verify your result (to within uncertainties associated with your measurements off of the diagram). Draw lines and add appropriate labels on your ST diagram to aid in the explanation of your answer.
(b) How long does it take Superman!to pass the car as measured by you? Answer this question as measured to scale on your ST diagram, and then do the calculation to verify your result (to within uncertainties associated with your measurements off of the diagram). Draw lines and add appropriate labels on your ST diagram to aid in the explanation of your answer.
(c) How long does it take Superman!to pass the car as measured by Superman! ? Answer this question as measured to scale on your ST diagram, and then do the calculation to verify your result (to within uncertainties associated with your measurements off of the diagram). Draw lines and add appropriate labels on your ST diagram to aid in the explanation of your answer.
Extra Problem 1.N The Pole/Barn Paradox (again!)
(a) A relativistic runner carries a 5.00-m pole straight toward the open front door of a barn. A farmer standing to the side sees the pole just fit into a 3.94-m deep barn. Show that the runner must be running with the speed 0.62 c.
(b) Let the barn and farmer be in the S reference frame, and the runner and pole be in the S’ frame. Determine an appropriate scale on your ST diagram and draw in the pole and its worldlines, and the barn and its worldlines. Draw the front end of the pole at the origin with its length extending along the negative x’-axis, and draw the barn with its front door at the origin with its depth extending along the positive x-axis, so that the front of the pole enters the front door of the barn at t = t’ = 0. (This will then make your diagram correspond to Fig. 1-37 on page 49 of your text—with different numbers, of course.) What is the paradox in this scenario? Discuss the pole-barn scenario as viewed from the farmer’s point of view and from the runner’s point of view, and address the paradox. (You need not extend your discussion to the case of a concrete slab at the back of the barn as was done in class.) Draw appropriate lines in your ST diagram and label appropriate events (as was done in the figure in the text) to supplement your discussions.
Extra Problem 1.O
An observer in S sees two explosions. The first explosion is seen at t1 = 3.00 s and position x1 = 1.00 km. The second explosion is seen 1.60 s later at the position x2 = 1.50 km. The reference frame S’ moves with a speed 0.600 c in the positive x direction as seen by S.
(a) What is the value of the spacetime interval (s)2 as measured in S? Is this interval space-like, time-like, or light-like? Why? Is it possible that the first explosion caused the second? Why or why not?
(b) At what time does an observer in S’ see the first explosion?
(c) How much later does S’ see the second explosion?
(d) What is the difference in position between the locations of the two explosions as seen in S’?
(e) Compute the value of the spacetime interval as seen in S’. Is this the value you expected to get? Why or why not?
(f) Does an observer in S’ measure the proper length between the two explosions? Explain your answer.
(g) What is the value of the proper length (or proper distance) between the two explosions? If it is measured by S’ (see your answer to part f) then you may just write down the answer (see your answer to part d). Otherwise call the corresponding reference frame S” and show how to get the proper length.
Extra Problem 1.P
A muon is created at some distance above the Earth’s surface (assumed flat), and travels vertically downward toward the Earth’s surface with a speed of 0.998 c as seen by an observer on Earth. (We will call this the positive-y direction.) The Earth observer measures the time between the muon’s creation and it striking a detector on the Earth’s surface as 28.1 s. At the instant the muon is created, a rocketship flies by the Earth with a speed of 0.850 c traveling parallel to the Earth’s surface and toward the right (call this the positive-x direction). The three objects (Earth, muon, rocketship) form three inertial reference frames with their axes parallel to one another; we let all reference frames detect the creation of the muon at t = 0, x = 0, and y = 0 (and with the associated primes for the other reference frames)—that is, all of the coordinate axes of the different reference frames line up at time equals zero.
(a) What are the spacetime coordinates (x,y,t) in the Earth (S) reference frame of event 1 when the muon is created, and event 2 when it collides with the Earth?
(b) What are the three coordinates of events 1 and 2 from the Muon’s reference frame (S’)?
(c) What are the three coordinates of events 1 and 2 from the rocketship’s reference frame (S’’)?
(d) With what velocity does a person on the rocketship see the muon moving? (Magnitide and direction relative to x’’-y’’ axes; as always, draw a sketch of the axes, the vector, and the angle!)
(e) What distance does a person on the rocketship see the muon travel between events 1 and 2?
(f) Compute the spacetime interval between the creation and detection of the muon in all three reference frames.
(g) Does there exist an inertial reference frame in which event 2 is seen to take place before event 1? If so, find one; if not, explain why not.
Extra Problem 2.A
Use work/energy arguments from Lecture 2.I to support the argument that no material object (non-zero rest mass) can be accelerated up to the speed of light in a vacuum.
Extra Problem 2.B
Prove that the expression for the relativistic kinetic energy becomes the classical expression in the limit of non-relativistic speeds.
Extra Problem 2.D
Prove that the magnitude-squared of any generic 3-D vector in real space (that is, a type of normal 3-D vector we’re used to working with) is invariant under a rotation by a general angle about the z-axis. (Do not do this for a specific vector for which you have made up numerical values for the components—you must do this for a generic vector. Just define symbols for the components in the non-rotated reference frame [for example, rx, ry, etc.], and then show that the magnitude-squared has the same value in the rotated system as in the non-rotated system.)
Extra Problem 2.E
Prove that the magnitude-squared of any generic 4-vector A is invariant. That is, prove that
for any general 4-vector. (Remember that a 4-vector is a mathematical beast whose components transform according to the 4-vector transformation equations. This is the definition of a 4-vector! [...except for maybe the “beast” part....])
Hint: You will not want to use the summation notation in your proof! This is very straight-forward, even if the equation above looks intimidating—just use the 4-vector transformation equations to complete the proof...