1. A light source is located at rest at the origin (x=0,y=0) of a coordinate system. It flashes at t = 0. The event (the flash) is defined by x=0, y=0, t=0. The light pulse expands in all directions (at the speed of light). At what time does it pass the point (x=300,000 km, y=0)? The point (x=0, y=600,000 km)? The point (x=300,000 km, y=600,000 km)?

How do these answers change if the light source has a speed of .5c when it flashes? Does it matter in which direction the light source moves?

2. A light year (ly) is the distance light travels in one year. A light second (l-sec) is the distance light travels in one second.

a. Find the number of meters and kilometers in each.

b. Convert the numbers given in problem 1 into light seconds and convince yourself that your answers still make sense and that the arithmetic is now much easier.

3. San Francisco and Los Angeles are separated by 400 miles. One car leaves San Francisco traveling south at 50 mph, while the other leaves Los Angles at the same time traveling north at 60 mph.

a. At what rate (miles per hour) does the separation between the two cars decrease? (Hint: how far apart are the two autos after the first hour?)

b. How far from San Francisco will the cars be when they meet?

c. If instead, both cars travel north from their starting points, at what rate does their separation change? Where do they meet?

4. Redo part c for the case where both cars travel south.

5. A rocket ship of length L (in the rocket's frame - its proper length) leaves the earth at speed V. A light signal is sent after it and arrives at the rocket's tail at t' = 0 (rocket time) ant at t = 0 (earth time).

Hint: drawing some pictures will help you in answering the following questions.

a. When does the signal reach the head of the rocket according to rocket clocks?

b. When does the signal reach the head of the rocket according to earth clocks?

The signal is reflected from the head back toward the tail

c. How long does it take to reach the tail of the ship according to rocket clocks? What do the rocket clocks now read?

d. How long does the return trip take according to earth clocks? What does the earth clock at the tail of the ship now read?

e. Take the ratio of the clock reading in b to that in a. Do the same for the readings in d and c. Make some sense of your answers.

7. Alpha Centauri is a star about 4 light years away. For a rocket to make the trip in one year (as reckoned by its occupants), how fast must it travel?

To the occupants of the rocket alpha-Centuauri would appear to be approaching at the same speed. How far away does it appear, to them, to be as they start the trip.

How fast should the rocket travel in order to make the trip in one day (rocket time)?

8. A muon (one of the unstable elementary particles) flies through the lab with speed v = .998c. The muon lives 31.6 x 10^-8 seconds as measured in the lab. How long does it live in its rest frame?

9. A 20 meter pole is carried so fast in the direction of its length that it appears to be only 10 m long in the "laboratory" frame of reference. Therefore, at some instant the pole can be entirely enclosed in a barn 10 m long. However, look at the same situation from the runner's reference frame: to her the barn appears to be contracted to half its (proper) length. How can a 20 m pole fit into a 5 m barn? Explain. Draw some pictures to show what everyone sees and when.

10. An observer reports that two missiles are moving parallel to one another on a straight line path, one with speed .9c and the other with speed .7c. Find the speed of one missile with respect to the other.

Consider two cases - both missiles moving in the same direction and the two missiles approaching each other.

11. Some practice with the Lorentz tranformation equations - using your answers to problem 5: Use the answers you obtained for parts a and c (rocket times) and the Lorentz equations to get the answers to parts b and d (earth times). You should get the same answers you did before.

13. The frequency of green light is about sqrt(2) times that of red light.

Find the speed of the alleged traffic violator, who in good conscience tells the judge that the top light on the signal was green. Is he guilty?

14. Observers on a train see a photon traveling vertically (in the y' direction). Find u_x and u_y as measured by station observers who see the train going by at speed v in the x direction.

Draw a vector indicating the direction of the photon's travel (its velocity).

Recall the situation: Ulysses travels to planet P (8 ly away) at v = .8c and then returns. Homer ages 20 years and Ulysses ages 12 years.

Each sends out radio signals or pulses at the rate of 12/yr (each determines his own rate). Each can then count the pulses received to "measure" the other's age.

1.Ulysses receives signals at the rate _____ /yr as he travels away for what he says is years, giving a total of _____ signals received.

2.On the return trip Ulysses receives signals at the rate of _____ /yr for years giving a total of _____ signals while returning.

3.The roundtrip total is ______ signals. He therefore concludes that Homer must have aged _____ years (Homer's time) while Ulysses says he has aged 12 years.

1.Homer receives _____ signals/yr as Ulysses goes away, toward P. The last of these "away" signals reaches Homer _____ years after Ulysses reaches P; therefore Homer receives "away" (redshifted) signals for _____ years. He receives a total of redshifted signals.

2.Homer then begins receiving "toward" (blueshifted) signals at the rate of

_____ /yr. He will receive these for only _____ years, giving a total of

_____ blueshifted signals.

3.Total number of signals received = ______ . Therefore Homer concludes: Ulysses ages ____ years (during Homer's 20 years).

Show that there are frames in which the temporal order of the events is reversed.

Can the temporal order to these events be reversed? Why is this a good thing?

1. Set up a Minkowski diagram using Homer's coodinates for the orthogonal axes.

Include on the diagram

a.Homer's worldline

b.Ulysses' (bent) worldline

Be sure to get the proportions and the angles correct (use a ruler) - you will need to be able to read values off of the axes.

2.Make three copies of your Minkowski diagram and use one for each of the following

I.Show Ulysses' space and time axes for both the outgoing and the return trips (label them t' and t'' etc.)

II.Draw the the worldlines of the (photon) signals sent out by Homer and received by Ulysses (to keep the diagram readable, let the emission rate be 1/year).

a.How many signals does Ulysses receive before changing direction to head home? Is this the rate expected from the Doppler effect?

b.How many signals does Ulysses receive on his return trip? Is this the expected rate?

III.Draw the worldlines of the signals sent out by Ulysses and received by Homer.

a.How many signals does Homer receive during the first half of his wait?

b.When (t=?) does Homer receive the signal sent out by Ulysses as Ulysses changes direction?

c.How many signals does Homer receive after the time found in b?

Does each of the twins receive the proper number of signals?

Is the Doppler effect apparent on your diagram?