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Special Relativity Mid-Term Exam

Due: Friday, November 8, 5 PM.Hand in: Andrew Brainerds office, Pupin 931

The exam comprises a number of problems, some on par with simple exercisesthat wont take you long to finish, some that are somewhat more involved.Do all problems. You can use any of the material from class but dont useanything else (you have all the material you need to solve all the problems). Nocollaboratingall work must be your own. This exam is as much for you as it isfor me: doing these problems will help solidify your understanding of everythingweve done in the course so far.

1. A long jump contestant wants to jump a distance L in the frame of reference

of the stationary judges. She knows she will stay in the air for a time T,measured in her frame of reference. How fast does she need to be goingwhen she jumps in order to do this?

2. Two rockets are located at a point between two planets, equidistant fromeach planet. The pilots decide to have a race each rocket will head towardsone of the planets and whoever arrives first wins. If the rockets travel atthe same speed, who will win according to (i) a stationary observer and(ii) each pilot, and why?

3. A muon exists for 2.2 microseconds (=2200 ns) before decaying in its restframe. If a muon travels 1100 before decaying, how fast was it traveling?

4. In one frame of reference, a rocket travels a 10 lt-yrs distance over 30 years.

In another frame of reference, the rocket has only traveled 5 lt-yrs. Whatis the time elapsed during the trip in the second frame of reference?

5. In the movie Planet of the Apes, an astronaut leaves the Earth in 1972,ages by 18 months on a spaceship and returns to Earth in the year 3978.If the spacecraft he was on was traveling at a constant speed (ignoringfor now the moment at which it turned around and headed back towardEarth, something we will return to in class shortly) how fast was it going?

6. A train passes by a platform, heading east. Both the platform and the trainsynchronize their clocks so that the clock at the center of the train and theclock at the center of the platform both read 0. According to observers onthe platform, the clock at the front of the train is behind by 100 ns. If the

Lorentz-contracted train is the same length as the platform and the rearof the train passes the eastern end of the platform at (platform) time t =200 ns, how long is the train?

7. In some reference frame F, event A has coordinates (t = 2, x = 5) andevent B has coordinates (t = 4, x = 4). Find a reference frame F in which

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events A and B are located at the same position x, and describe how F ismoving with respect to F.

8. A rocket leaves Earth at a velocity (3/5)c. Ten years after leaving, aceremony is held on Earth to commemorate the rocket launch. Accordingto clocks on the rocket, when do the rocket crew first see this event(pre-processing), and when do they say it occurred post-processing?

9. Two planets are located 10 lightyears apart. One planet sends a light signalto to the other planet, which the other planet then reflects back. At thesame time as the first planet sends the signal, a rocket leaves the firstplanet headed for the second planet traveling at 0.5c. Make a spacetimediagram in the frame of the rocket showing (i) the signal leaving the firstplanet (ii) the signal being reflected at the second planet, and (iii) the firstplanet receiving the reflected signal. Label each event with coordinates

for both the rockets frame of reference and the first planets frame ofreference.

10. Consider two broomsticks whose rest lengths are each 255, with the onlydifference between them being that one is red and the other is blue. Fromyour perspective on Earth, the blue broomstick is stationary, while theother red broomstick is rushing by in the positive x-direction with speedv = 8/17c.

a) From your perspective, what is the length of the red broomstick, andfrom the perspective the observers riding the red broomstick, how long isyour blue broomstick?

b) Each broomstick is equipped with a light on each of its ends. Observersthat are riding the red broomstick flash both of its lights simultaneously,from their perspective. How far apart do you say the flashes took place?

c) You flash the lights on your blue broomstick simultaneously. Accordingto those riding the red broomstick, how far apart did these flashes takeplace?

d) As the team of observes riding along with the red broomstick pass byyou, they grab your blue broomstick in a manner thats simultaneous fromtheir perspective. When they subsequently compare the length of the redand blue broomsticks, will they say that the blue one is longer than thered, shorter than red, or equal to red?

e) What speed must a third frame of reference have, from your perspective,in order that observers in this new frame claim that both broomsticks havethe same length?

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f) Show that your answer reduces to the Newtonian answer in the limit ofsmallv .

11. Imagine now that a team of observers (we will call them Team Red) areriding the red broomstick, and zip by Earth at speed v . Assume that theorigins of Team Earth and Team Red cross each other at t = t = 0(thatis, we are using the usual convention that (t= 0, x= 0) = (t = 0, x = 0)).Now imagine that you learn that one member of Team Redthe one attheir originhas stolen a top secret file that he intends to sell inhabitantsof the distant planet Zaxtar. So, upstanding citizen that you are, you

jump on your new super broomstick, and chase after Team Red at speedVwhich is larger than v . You leave Earth at timet = t1. To distinguishthe various values ofthat come into this story, lets write [w] = 1

1w2for any speed w (expressed in units wherec = 1).

a) At what time t2 according to Earth clocks do you catch up with theoutlaw in Team Red?

b) What is the time according to Team Red when you start chasing afterthem?

c) According to Team Red, how far away are you from the outlaw whenyou start chasing them?

d) According to Team Red, at what time do you catch up with the outlaw?

e) According to Team Red, how long were you chasing them before youcaught up to the outlaw?

f) According to your watch (which read t1 when you left earth), whattime is it when you catch up with the outlaw? (Dont concern yourselfwith the affects of acceleration. Instead, if you are bothered, imaginethat the question is phrased more precisely as: Another observer movingpast Earth with speed V , passes the origin of the Earths frame at timet1 on their own clock (and on Team Earths clock). What time will thatobserver say it is when they catch up with the outlaw who is at the origin ofTeam Red?) Please solve this two ways, one of which involves the LorentzTransformation between the your new frame (that has speed V relative toEarth), and Team Red (which has speed v relative to Earth) and show allthe relevant features in a space-time diagram.

12. Imagine you have two bicycle wheels whose hubs are both welded to ametal shaft, with rest length L0. Assume the shaft is aligned along the

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x-coordinate axis so that the wheels can spin about that axis. A red dot ispainted at one location on the left wheel and a blue dot is painted at the

corresponding location on the right wheel (so that the (y, z)coordinates ofthe two dots in an (x,y,z) coordinate system agree). As the shaft spins inthe laboratory with angular velocity, the red and blue dots on the wheelsare aligned with each other according to an observer at rest with respectto the contraption. Now imagine that your friend zips by the laboratory,headed in the positive x-direction, at a speed v. Will your friend agreethat the red and blue dots are aligned? If not, what will your friend claimto be their angular offset?

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