Prove that for a spacelike interval, two events cannot occur at the same place in space

Observer A, who is at rest in the laboratory, is studying a particle that is moving through the laboratory at a speed of 0.624c and determines its lifetime to be 159 ns. Observer A places markers in the laboratory at the locations where the particle is produced and where it decays. How far apart are those markers in the laboratory?

The center of our Milky Way galaxy is about 23000 ly away. (a) To eight significant figures, at what constant speed parameter would you need to travel exactly 23000 ly (measured in the Galaxy frame) in exactly 25 y (measured in your frame)? (b) Measured in your frame and in lightyears, what length of the Galaxy would pass by you during the trip?

Recall, from this chapter, that the factor gamma (γ) governs both time dilation and length contraction, where When you multiply the time in a moving frame by γ, you get the longer (dilated) time in your fixed fame. When you divide the length in a moving frame by γ, you get the shorter (contracted) length in your fixed frame. Assume that rocket taxis of the future move about the solar system at half the speed of light. For a 1-hour trip as measured by a clock in the taxi, a driver is paid 10 stellars. The taxi-driver’s union demands that pay be based on Earth time instead of taxi time. If their demand is met, show that the new payment for the same trip would be 11.5 stellars.

As seen by one observer, two firecrackers are set off at the same place but separated by 4 seconds in time. As observed by a second observer, moving with respect to the first, the two firecrackers detonate at places separated by 3 light-seconds in space. As observed by this second observer, how much time is there between the two detonations?

A laser beam moving at velocity c relative to us is turned on just as three (hypothetical) observers pass by us in the same direction as the beam. Observer 1 has velocity c/2, observer 2 has velocity 0.999c, and observer 3 has velocity c. If metersticks had different lengths in different frames, or clocks ticked at different rates in different frames, it is not impossible that the first and second observers could measure the beam to be moving at speed c relative to them, but the third observer could not find that the beam is moving at speed c relative to her. Prove this quantitavely.

The crew of an enemy spacecraft attempts to escape from your spacecraft by moving away from you at 0.283 of the speed of light. But all is not lost! You launch a space torpedo toward the foe at 0.351 of the speed of light with respect to you. (a) at what speed in kilometers per second does the enemy crew observe the torpedo approaching its spacecraft? (b) Is this more or less than the classical limit? Use the Galilean transform to prove this. (c) What if the torpedo is launched at the speed of light? At what speed in kilometers per second does the enemy crew observe the torpedo approaching its spacecraft? (Show all work.) (d) How fast would the second craft have to be going to measure the torpedoes speed as 10% greater than the classical limit. (Assume the torpedo is launched at the original speed, 0.351 of the speed of light.)

Choose the option that makes the following statement cor- rect. Two events at a single location define a time interval. The proper time interval At, is measured by an observer [(a) at rest; (b) moving] relative to the location where the two events Occur.

In the Marvel comics universe, Quicksilver is awfully fast. Let's say he can run at a velocity of 0.56c. He measures a trip as having a distance of 4.19e+05 m. How much time does Quicksilver measure this trip as taking? Does Quicksilver measure the proper or dilated time? Quicksilver's sister, Wanda Maximoff, is standing stationary near where he is running. How much time does Wanda measure this trip as taking? What distance does Wanda measúre for Quicksilver's trip?

According to observer O, a blue flash occurs at ?1 = 10.4m when ?1 = 0.124µs and a red flash occurs at ?2 = 23.6m when ?2 = 0.138µs. According to observer O’, who is in motion relative to O at velocity ?, the two flashes appear to be simultaneous. Calculate the velocity ?.