More Sample Problems
1. The day that Eugene is born, his father reads in the local paper that there is a mysterious spaceship, now 10 light-years away and heading towards earth at 0.8 c. Always an optimist, Dad hopes that when the spaceship gets to earth, it will be bringing a playmate for Euguene, a kid who will be exactly Eugene's age when he arrives.
(a) How old will Eugene be when his new friend arrives?
(b) According to Eugene's parents, when was Eugene's new friend born?
(c) Before the spaceship slows down, when, according to Eugene's new friend, was Eugene born?
2. Angered by the despicable behavior of some beast at a distance L from you, you whip out your tachyon gun and shoot it dead. Assuming your tachyon bullets travel at 2.5 c, calculate the interval between the two events: (A) You shoot, and (B) The bullet hits the beast. Then comment on the implications for causality.
3. Consider the clock-synchronization problem from the first actual test. In class today, we calculated the time interval (in the train's frame) between the two events (clock A starts, clock B starts).
(a) What is the time interval between these events in the station's frame? Is either time interval a proper time?
(b) Find the positions of the two events in the train's frame. What is the spatial separation between the two events?
(c) Calculate the interval between the two events in each frame. How do the intervals compare with each other? Are they spacelike or timelike, and is this what you would expect?
4. A suspicious husband hires the With-it Detective Agency to spy on his wife. They report that on Wednesday afternoon, his wife and the good-looking guy who drives the UPS truck in their neighborhood were both on the Occident Express train at the same time. When her husband confronts her with this information, accompanied by insinuations about what might have been going on, the wife hires Too-Cool Investigations to save her reputation. They interview the train employees, who resolutely deny that the two were on the train at the same time, although they do report that each was on the train at some point in the afternoon. Is it possible that everyone is right?
5. Suppose that when a particular heavy nucleus decays, it emits an electron that moves away at a speed of 0.8c relative to the nucleus. Furthermore, suppose that such a nucleus is moving relative to us at 0.6c. Relative to us, what is the highest speed we would measure for the electron? What is the lowest speed we would measure?
6. At 5 pm the northbound Wonderland Express leaves the Center City station, with Mrs McGillicudy aboard. After she reads the comics and completes the crossword puzzle, she happens to look out her window and, to her horror, sees someone being strangled on one of the southbound locals. Her watch reads 5:30. She reports this to the train police, who assure her that they will be able to identify the criminal, because the stationmaster keeps a record of who is on each of the local trains. All they need to do is to determine when the relevant train ought to reach the station, and they'll know who their suspects are. If the express travels at 7 MPH and the locals travel at 5 MPH, when will the criminal's train reach the Center City station?
7. Here on earth we suddenly notice that we're seeing an asteroid heading our way.. The signal from the asteroid has just now (let's say this is t=0) arrived, and from its redshift we conclude that its gamma relative to us is 8. Also, from triangulation, we conclude that this signal left the asteroid when it was 50 light-years away.
(a) How fast is the asteroid travelling, and where is it now?
(b) Because it's coming straight at us, we decide to blast it apart. We immediately send a mighty laser pulse at it. Where and when, according to us, does the laser pulse reach the asteroid?
(c) The laser pulse just misses the asteroid, but a little scattered light does wake up some aliens hitching a ride on the asteroid. How long, according to them, do they have to make plans before the asteroid collides with that cute blue planet?