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At points A and B on Earth, distant at a distance of l = 10 km,two events took place simultaneously, for example, TV screens werelit. The number of microseconds sep...

Question

At points A and B on Earth, distant at a distance of l = 10 km,two events took place simultaneously, for example, TV screens werelit. The number of microseconds separating these events from thepoint of view of an observer on a spacecraft moving away from theEarth along the straight line AB at a speed of v = 0.6 s, where cis the speed of light, is

At points A and B on Earth, distant at a distance of l = 10 km, two events took place simultaneously, for example, TV screens were lit. The number of microseconds separating these events from the point of view of an observer on a spacecraft moving away from the Earth along the straight line AB at a speed of v = 0.6 s, where c is the speed of light, is



Answers

A spaceship travels at $0.80 c$ from Earth to a star 10 light years distant, as measured in the Earth-star reference frame. Let event A be the ship's departure from Earth and event B its arrival at the star. (a) Find the distance and time between the two events in the Earth-star frame. (b) Repeat for the ship's frame. (Hint: The distance in the ship frame is the distance an observer has to move with respect to that frame to be at both events-not the same as the Lorentz-contracted distance between Earth and star.) (c) Compute the square of the spacetime interval in both frames to show explicitly that it's invariant.

So this is the information we have. So the distance between the Earth and the sun is 8.33 light minutes and we know that relative to the Earth and the sun we have this information for events A and B. So we go ahead and we calculate for our first velocity for our first reference frame that we're gonna look at, We write down the velocity, we calculate Gamma, we have 1.512 So now we're going to calculate the time for event a relative to this reference frame, we call that ta prime, go ahead and substitute. Using the Lorentz transformation, we have zero minus 00.75 c overseas square time, zero in parentheses. That gives us zero also for ta prime. We go ahead, we calculate for TB prime, and we substitute in our values. For this time we end up with a negative value. We get negative 5.74 minutes. This tells us actually, that event be happened before event a relative to this reference frame. Unlike in the earth's sun frame, where event a happened before event be. Also, there's a longer time interval between these two events, unlike in the Earth's sun frame. So it's a reverse order and a longer time interval. Now we go onto the next reference frame. So instead of a velocity of 0.75 c, we now have a velocity of negative 0.75 c. So we go ahead and would calculate gamma, which is the same value as before. And now we get to see a prime and TV prime. But this time we have to use the inverse Lawrence transformation with a positive sign because the velocity is in the other direction. So again, tea a prime is zero. Let me scroll down and we're gonna get TV prime, and we see that it's 13.15 minutes. So it's a positive value for TV prime, and we get the same order of events as we do in the Earth's sun frame. But we have a longer time interval. Now we go onto the frame, uh, moving down. Here we go onto the frame where the velocity is 0.294 c relative to the Earth sun reference frame. Forget gamma, and we calculate our times. Once again we get zero for Ta Prime and we substitute the numbers and get TV prime, and we also get zero. So this tells us that if both times or zero they happen at the same time, so we have simultaneous events in this reference frame.

All right. So for question A. Well, we know that the spaceship who already travel distance that's a do you want. Okay, which is simply go to B times too. You want OK? V is given as on through pointing a zero C, which is 0.0 times three times 10 to about a meter per second, which will give us the speed of this patient is it's about 2.4 times on attention about eight meter of a second. And we know t one was given as one point. Oh times 10 to the power. Four seconds. So therefore, before the signal was sent out, the special traveled this and the one which is equal to oh, 2.4 sighs tend to power a. You have a second and in times one point no times tend to Paula for Sacco's. And this will give us 2.4 as central power off 12 meter. Okay, so you wanted to determine how far should signal travel. But when you to first determine that, um, the time first we know that if we assume that a decision that a signal travel is equal D therefore, you should be good to the speed of signal which is the speed of light ties. A tiny who takes this ace ti Okay, it's therefore we know a T. Can you go to the distance? Do you have about a speed of signal to see? Okay, so for this time here on the spaceship who also traveled this much time before the signal catching up, Okay, Therefore, the total distance, which is simply you go to we times t plus do you want OK? And when the tea is it was a d oversee every we have three times the oversea plus do you want and then we'll have d is equal to we oversee times D plus do you want? And when OVI was just zero point a zero c and do I see a tent city? We don't know. Ok, and when I d want is, um 2.4 times essential power of 12 meter. As you can tell here C and C coming, Cancellara, if we move the zero point Israel d to the left I who have t minus 0.80 the which is your 0.20 p. Yeah, and this is equal to 2.4 times 10th. Willpower. 12 meter. And this will give us the distance. That is sick. No need to. Travel is about 12 times 10 to the power of 12 meter, which is 1.2 times essential. Apollo 13 meter. Okay. And for the next question, he was asking us the total time there was a suspect and to be spent. Okay, so that's a teetotal. So we know Tito, welcome. Even through the time from the Greek arrangement was just the one and plus the time for the signal to reach the spaceship, which is t okay. And we know T. Can we you go to, um, the distance that a signal travel over the speed off the signal, which is C. Okay. And this will give us 1.2 times 10 to the power of 13 meter over three signs. Send to a borrow eight peter per second, and this will give us four times tend to about four seconds. Okay. We already know to you want which is the time for the pre arrangement. Just about no one point no times. 10 to the power. Four seconds. So therefore we know the total time will be you go to a said he total is equal to one times 10 to Paul. Four seconds plus four times 10 to about four seconds. And this will give us a toast. Times well, by times tend to Apollo for seconds. Okay. And just my answer for this question. Thank you.

We begin discretion by solving expression for the time dilation to develop city, so these goes as follows. Delta T invited by Delta to use the rope is the coast one divided by the square root of one miner's be divided by sea squared. Now we can feed both sides to get noticed. The hero divided by Dougherty is it goes to the squarish off one miner's Be divided by C Square. No, we can square both sides to get No. Zero. Divided by the upper teeth. Square is equal to one. Miners will be divided by C squared. Finally sent this term to the left and disturbed to the right to get the divided by C squared is opposed to one minus delta t zero divided by Dr T squared And think this square root so many divided by sea you see close to the square it off one miners. Now that the zero divided by Delta T Square Now we have to plug in the values that were given in the problem to get the wrench here between B and C to do it. Remember that Delta T zero is the proper time on Delta. T is a violated time in this question that the events that we're measuring are the picks off the clock inside the spaceship. So the proper time is one second in the deleted time is one minute. Therefore we have the following is equal to the square root off one minus. Not that easy, you know, which is the proper time. So one second decided, like in a little time off one minutes. So we have 60 seconds squared these gigs this V divided by sea approximately 0.999864 The velocity. The relative velocity off the spaceship is equal to 0.99986 times, see?

Okay, so in front as prime will have the position acts in as prime is able to x minus VT over square rule Ministry sport of C squared. So when I only ask value in, the friend asked. You see, with the 150 killed year, which is 1.5 times in five years and whenever velocities given 0.6 C, which is 1.8 constant call a meter per second and when the time is given, four point vitamins in departing four seconds. So now if you're playing better, the question we can determine the exposition in front s prime, which is equal to own one point by times, tend to qualify meter minus ongoing. A times tend to power a meter of a second 10 times 4.5 times 10 to the power of 94 seconds over square root, one minus zero point six C to the power to oversee square. And this will give us the exposition in the front, as prime is equal to, uh, a pointing. 6 to 5 times tend to the power of four meter if we come worried. Two kilometer is about 86.25 kilometer. So we know since, uh, he moves along the share as direction. That means Y prying and Z primes. You have the same value from the wide see value in the front s so have ah, wire finds It was why I was able to 15 kilometer and z prime is he couldn't see which is equal to 1.0 kilometer. So now that's determine the time in the front s trial, which is you through T minus V s overseas square and then also scored one minus weeks. World sees work. So if you're talking about who Spain to the equation you have t prime is equal to four point by times 10 to the power of number four seconds minus 1.8 times 10 to the power of a meter per second 10 times one point by times 10 to the power of five meter by meter and it over three times Tend to a power a meter per second to a power to because we know, see isa spiel and speed. A lot is equal to, um three times tend to power a meter per second and this holding here over square root one minus zero. Going six c toe a power to oversea square and it will give us, uh, the time in the front s prime is about one boy a 75 times, 10 to the power off 94 seconds. So now we know the flash who occur in ah s prime assistant. When the exposition is equal to 86.2. Vacuum eater and wipe Ryan's with 15 kilometer, you probably go along, kill Amir and the times is equal to 1.87 Fighters tend to power in 84 seconds and these are the answers for this question.


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