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Leneth Keniclo Hizelime 060 1747 4 "ncn 5lnIonary Question (Z0 pts) An clencntary Yoclo 0[ 0950: rcletive i0 & Iaboraloty fefcrencs mst frame. If this particlc has lifeluno nasuned in dhat iratE uing Acmc hor Iar cal Aruvcl [rnmic claseical (Ehyaica? (0) acccnling ul [ tur obzcrvalion? acecnling sreciul relativity? peun Idopen Ihar Wc #ave [uncliun

leneth Keniclo Hizelime 060 1747 4 "ncn 5lnIonary Question (Z0 pts) An clencntary Yoclo 0[ 0950: rcletive i0 & Iaboraloty fefcrencs mst frame. If this particlc has lifeluno nasuned in dhat iratE uing Acmc hor Iar cal Aruvcl [rnmic claseical (Ehyaica? (0) acccnling ul [ tur obzcrvalion? acecnling sreciul relativity? peun Idopen Ihar Wc #ave [uncliun



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Suppose $x(t)$ is the locus of points in the frame $K$ at which the readings of the clocks of both reference system are permanently identical, then by Lorentz transformation $$ \begin{gathered} t^{\prime}=\frac{1}{\sqrt{1-V^{2} / c^{2}}}\left(t-\frac{V x(t)}{c^{2}}\right)=t \\ \text { So differentiating } x(t)=\frac{c^{2}}{V}\left(1-\sqrt{1-\frac{V^{2}}{c^{2}}}\right)=\frac{c}{\beta}\left(1-\sqrt{1-\beta^{2}}\right), \beta=\frac{V}{c} \end{gathered} $$ Let $\quad \beta=\tan h \theta, 0 \leq \theta<\infty$, Then $$ \begin{gathered} x(t)=\frac{c}{\tan h \theta}\left(1-\sqrt{1-\tan h^{2} \theta}\right)=c \frac{\cos h \theta}{\sin h \theta}\left(1-\frac{1}{\cos h \theta}\right) \\ =c \frac{\cos h \theta-1}{\sin h \theta}=c \sqrt{\frac{\cos h \theta-1}{\cos h \theta+1}}=c \tan h \frac{\theta}{2} \leq v \end{gathered} $$ $(\tan h \theta$ is a monotonically increasing function of $\theta)$

What a healthy given problem. We use the Lorentz transformation equation to ride to Delta T dish. Is it called a Gamma dies Delta two minus three times Delta X, divided by C Square And this will be this will be a gamma times Delta T minus a better you, don't we? And seeing terms off a better speak Very little Delta X divided by sea. Some student values we get to your gamma times Delta to you Jeez, one times 10 to the power minus six seconds minus a better times Delta X into the 400 meter are divided by the speed of light, which is three times Tintin politi meter per second Where the Lawrence factory. So, um, there's a functional for better. Um, we get, uh, so this is an expression for Delta T dish for a party or about to be a plot off Delta T is a function off. Better as you can see from this suppression in a range off or better, better greater than zero and less than zero for 01 We converted this plot. Are you following me? So let's say on X axis were Betta. So this is 0.1 This is a 10 and that this is the one. This isn't a delta, the delta t dish. Then we see ah, equation Reliable beats strange constant up to zero point 01 We know that the limits off the world relaxes are so here will be, uh, plus two micro second been here this day here in opposite site. This will be a minus two microsecond. We also know that how flayed the curve he's in this graph. The reason is that for low values off better moreover, aliens off the better. Ah, bull vehicles measure that temporal or measure off The temporal suppression between the events is approximately is ours. Uh, which is nearly one microseconds. Ah, there are no known in teaching on electricity effects in this place. Um, when re Plaut the delta t. So when re plot to delta t here. So let's say a year care. Now we're better up. And here is on a delta T dish for the values which are 0.1 uh, the 0.1 better greater than gentle 0.1 in less than one. We see our from this is ill and here it won't. He's a little color from Globo. Nicky cross here and a little flat and down. So the point it crosses here is Ah, a round of 0.8. So here is the one hand this is zero. Uh, here's one billion years over to microsecond. Must Michael a beer is our minus two microseconds. But to deal with the problem setting our So this was for C for a part to d d Don't setting delta today. She's a cool toe a gamma into delta T minus Betta delta X divided by sea. Um, from here, substituting your values for our gamma into one times 10 to the power minus six seconds minus better 400 meters, divided by speed of light three times 10 to the power a T two per second is it will be zero setting. This is a court zero. Oh, uses a better value, which is sea Delta T. Bye bye. Delta X subsidiary radius. We get here approximately, uh, approximately zero corn 750 But he off. The problem from the a graph that be shown in the party has increased the speed. The temple suppression, according to Bull winkles, is a past you for the lower values and then goes to zero. And finally, is the speed approaches. The speed of light becomes progressively more vanity. Yes, we can see here, So let's speed increases. It becomes more negative here for lower speeds. Lower speeds with Delta d dish greater than zero. Ah, we've been right. Delta Tiu A dish is less than TV dish, which implies there Oh uh, better less than 0.750 according to the bull vehicles, even a occurs before even be. Is he just off, though, or about F for the higher speed? A Delta T dish is less than zero. So for this, this teed ta dish will be greater than T B dish. That is Ah, 0.75 zero are less than a better, which is a less than one, according to Bullwinkle, even to be awkward before even a opposite off what we observe for a part of G. Ah, the answer is no. Even a cannot calls even to be or the wife's more. So we noted that the Delta X, divided by Delta T, is 400 meter divided by one microsecond, and this gives us four times 10 to the power eight meters per second, which is greater than which is greater than the speed of light. The signal cannot travel from even today to be without exceeding the speed of life. So the causal influences cannot. Forties Nate, it's a and thus they affect what happens to be your wife.

Mhm. Hi friends at the strength of pictures is taken at the instant. The picture is taken the coordinates of A. B. Eight as and be death in the rest frame of A. B. R. A juro juro juro juro. The zero and not 00. Did as 0000. Yeah it is having the coordinate one minus and not. Yeah. And to root off one minus B squared by C square judo judo. Mhm. In the stream coordinates of Vidas at other times. Oh leaders P beauty judo judo. So the dress is a project to be at the time. G. It's called to and not baby in in the frame in which be death addis is at rest that time cause corresponding to time corresponding this sorry time corresponding this it's white Lauren hodge transformation. Do you not? We just will be blown apart Route of 1- We square by C sq into and not by the when SB and not upon C. Square. So it could be written us and not by B into one minus. Okay. The square by C squared similarly in the rest frame of eight maybe to coordinate of a at other time. Mhm. Yeah. Uh huh. Hey that's the minus and not. And the root of one minus B squared by C square. Bless BG comma zero comma zero A. Death is a budget to a. Okay. At the time and not by we and the root of one minus B squared equals C squared the corresponding time. Okay in the frame encourage a death beaters are at rest is th S is equal to come on time. Steve A. And that is, and not by big that so thanks for watching it.

In this problem. We have a ship traveling with the unknown speed. Be relative to the there we're told the rest lengths Of the ship is 100 m. We'll rest links proper thing. Same thing. Yeah, length is in the frame that moves with that. As a test the ship which is a ship frame. So this is well that's if you like this or not, You go 100 m. That's the proper length. So lab frame is seeing a moving ship. So they will see contracted length something less. So now the first Now they've told us that Roosevelt left frame that they see the ship go by and they time how long it takes for the ship to go by. And they give the time is .2 microseconds. So they want to know what the speed is. Well, the time would be the length of the ship over the speed the ship is traveling. But this is not. No, not they're seeing a contraction length. This is the L not square root one. Man's B. Squared. Or is he squared or V. So what we gotta do now is some algebra. So let's solve for V. Well, we have to vis so we're not done. No, not over delta T square root. One might see squared over C squared. So we have that a square both sides. He squared. I'm not squared. That's a T squared. What must be squared. Of course he squared and bring all the V. Squared terms to the left. He squared one plus. Oh not squared. Over delta T squared. He could tell, not squared. Whoever does. T squared. Yeah, we can sovereign V squared. I'm not squared. Judge T squared one over one plus. L not squared. Got T squared. Yeah. Frost. This There should be a C squared here. Here. She squared. After we have and we can do a little more algebra. So we don't have to we don't have to have the al nahda and delta T. Multiple times went over. We can bring that inside is doug T squared over. L not square the denominator. So I'm going to get got the T or L not squared. And then when you flip this over and you bring it inside T squared on, it's going to knock out this. Tell G squared not square is going to knock out this house, not squared in the bottom. So this is just going to become one overseas squares. So we get from this. V is equal to one over. What I just wrote of tea over, L not squared was run over C squared have power so we can now put in our values. Talk to me like I said, is two microseconds. 0.2 microseconds I should say 3.2 times 10 minutes six seconds. They Rest length was 100 m square there plus one over three times 10 to the 8th meters per second. Square that and take the square root of all this. And this works out to be 2.6 Times 10 to the age meters per second. So that is the speed and part B. I want to know what the contracted like this and that is what we've already used. The al not described one by the square of a C squared. Well, you could just multiply the time Doctor T. Also. Uh huh. The only advantage of this revenge of this is that you may um This involved obviously there's always going to be rounding somewhere other. So if this would probably seen as the more precise got in the end. I really so about the same. Either you get it, if you use a 2.6 to multiply by the delta T. And get the answer for V. Or the 2.6, which you say is a rounded number sits in here squared. It's all about the same. Okay, so we got a 100 m Square root 1 -1 2.6 times 10 of the eight. Here's per second Over three times 10 to the eight. It is per second square that. And this works out to be 50 m so half. Yeah, for the rest length is a contracted length at a very high speed 2.6 2.6. Mhm. You're looking at In a .8 87 C. Units. Such high speed. You're you still have significant lengths even though it's half, it's still significance. Not like you're down to you know, 10th of a meter or something of that nature. Yeah. So that is the whole problem.

Our question says reference frame. It's prime. Moved a speed V equals 0.92 times the speed of light. See on Ben. This is a neat plus X direction with respect to the three origin of s and s prime overlap at T equals T prime equals zero. An object and station s stationary in the s prime frame at position X prime equals 100 meters. What is the position of the object and s when the clock and s reads 1.0 microseconds, So microseconds is 10 to the minus six. Okay, Right. T, because it's in the clock frame is 1.0 times 10 to the minus six seconds. And then do this according to the Galilean. And then the little rents transformation equations. So we're gonna do it classically. First, according to the Galilean, um, Ms says that this is part eh? X is just going to be equal to ex crime. The starting position, plus the velocity times the time. So this is going to be equal to s. So what you're gonna do is you're gonna plug in 1.0 meters for ex prime. You're gonna plug in 0.92 times C, which is 3.0 times 10 to the eight meters per second and the multiply that by our time, 1.0 types of minus six seconds. Plug all that into this expression and this comes out to be 300 in 76 meters. Okay, so that's using down lane transformations. So now let's use the Lorentz transformation. Well, Lawrence transformation says X is equal to gamma times. Ex prime plus v times t prime. Okay, well, we don't know t prime, we know T, but tea is related through t prime through this equation here, T is equal to gamma times t prime plus be of ex prime overseas for so now we just have to solve for t prime so we can plug that into our original equation. Our first equation, which I'll just little Red Star. Okay, so to do that, we can divide both sides by gamma. So this will be t divided by gamma and then subtract from both sides of the quotation V X prime over C squared. So Well, let's just, uh, just rewrite that, so we'll subtract from both sides. of the equation now. V from ex crime over C squared. So then this is t prime is equal to t over gamma A minus B of ex crime oversee square. Okay, so it will, just for all this. So these equations will get confused with one another. So now we can plug that into our expression for X X is equal to gamma. Gamma is one over the square root of one minus D squared over C squared times, ex prime. Close the times t prime. Okay, so we can plug in the values for for T prime, which we just found, which is t divided by gamma minus V of X crime divided by C square. Okay, All right, well, if you plug in all the values that we have for this expression which we have V c X prime t all written above. And don't forget that this gamma factor here is also won over minus V squared over c squared. So when over, gamma is equal to the square root of one minus Avi's, we're so that one minus b squared over the seas for comes to the numerator. So don't forget to change that There you plug all of that interview on this expression equals 316 meters. We can box set in as our solution.


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