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If a m 84.3 kg person were traveling 0.994c , where € Is the sppecd of light; what would be the ratio of the person'$ rclativistic kinctic cnergy the pen...

Question

If a m 84.3 kg person were traveling 0.994c , where € Is the sppecd of light; what would be the ratio of the person'$ rclativistic kinctic cnergy the penon'$ classical kinetic engrgy?kinctic energy ratio;What thc ratio of the person' relativistic momentum t0 thc person"$ classical momcntum?momcnlum rlio;

If a m 84.3 kg person were traveling 0.994c , where € Is the sppecd of light; what would be the ratio of the person'$ rclativistic kinctic cnergy the penon'$ classical kinetic engrgy? kinctic energy ratio; What thc ratio of the person' relativistic momentum t0 thc person"$ classical momcntum? momcnlum rlio;



Answers

(a) Calculate the relativistic kinetic energy of a $1000-\mathrm{kg}$ car moving at 30.0 $\mathrm{m} / \mathrm{s}$ if the speed of light were only 45.0 $\mathrm{m} /$ s. (b) Find the ratio of the relativistic kinetic energy to classical.

Now in this problem, it is given when a college excellent later between the right and we energy in by left front it is 6.3. No question is given it a proton is to be accelerated from you to it. Then how much kind of dignity get by the garden energy. Get my photo that we have to calculate in the first part and second part we have to calculate the ratio off Better city or collect point the velocity off road when they are released. Energy gin by the charged particle do toe electrically is defined it charge into potential difference energy gained by you Do their duty on this batteries given I energy gigabyte Good. You do there w so in a chicken by the pro Tony does so 6.3 So eloquent You know a weapon which the boat will get the same kind of dignity? No. In second part, we have to calculate the issue of publicity calculation for sickle part by electric energy. Skeletal happened Be swift, the violence, it even we So I stopped kind of dignity upon in well, a sitio electrode Marcel Collect put since both are getting the same kind of dignity. The velocity gained by the court on its my support. Both are getting the same kind of energy. If we divide equation one vato we will get then you simply by you will get myself Proto Marcel fled abroad on simplifying this vulnerability. The issue off Felicity off electron velocity or broken will be. What did you point here? So this is the answer up. Second part.

Okay. In this problem, we have a Compton scattering event. We're an incoming photons scatters from Elektronik Mass and is deflected at a ninety degree angle. The Lambda the wavelength of our photon is four point five pico meters. Either the negative twelve meters were asked to find a number of things. We have to find the kinetic energy of the electron after the scattering, how this kinetic energy compares to its rest energy M C squared rats for the velocity electron, the momentum of the electron and momentum relative to its non relativistic value. So this is heavily implying that the scattered electrons going tohave, relativistic, allot a relativistic velocity. So we'll have to be careful there were going to use the Compton scattering equation where which relates the change in wavelength to a scattering angle. We're gonna need to know the elect the energy of a photons as it's related to its wavelength. And we're gonna need to know the relativistic mo mentum, which involves Thie Lawrence Factor gamma, which we define as you'd find here. It's related to the velocity the object will be using place constant speed of light and the Mass one electron. Let's jump into it. So to start out, we need to find to find the kinetic energy of the electron. We need to know the change in wavelength of the photons in the scattering, so you can start by using the contents. Scattered equation. Why primary schools? Or plan to prime Michel's Lambda plus H of Ramsey on my ass co sign of ninety degrees because the photons scattered ninety degrees. As per the problem. So we find Sensei degrees is zero. This is really just clammed up, plus h over emcee h over M C is what is called the Compton wavelength, and we know it, and it's useful that perhaps have its value recorded because it's used frequently. So plug in or value for the initial wave length of our light plus our content wavelength, which is about two point four to six become years. Might think I'm using the wrong wavelength. That's right. I, uh, used on wrong wavelength here. Four point five zero nine twelve bust out two point four. United twelve gives us a roll result of about six point nine six point ninety six commuters. So this is the new wavelength of our life. So now that we know this new wavelength We can set up conservation of energy. So initially, the photons coming out arrest and arrest electron. So it'll have an energy of AIDS she ever Landa After the scattering, this photo will have a new anda. So the new energy and the photons or the electron rather will have a Connecticut or some connive and decay Consol for Kay by simply getting the change in the photon energy. Make it as easy as possible. Thank you. Now we can simply plug in are two different wavelengths. The initial on the final restart calculated We get a result of one point five three six times ten to the negative fourteen. Jules. And if you're curious, this is about ninety six thousand electron volts R ninety six k v But we don't need to use an electron volts. So now we have the kinetic energy We can compare it to its right rest Energy. Well, first, let's box this because something we we're looking to solve for so that as a ratio we can say comparing it to its rest. Energy rest. We have one point five either negative fourteen Jules compared to emcee which is well, M C squared rather nine point one, either negative thirty one kilograms times speed of light which is three times ten to the eight meters per seconds. No. And after performing this computation, we get a result of point one eight seven. So this this electron is is that about twenty percent the kinetic energy of light, which is pretty considerable. So this is something we need to be looking out for him. The rest of the problem that this definitely is a relativistic Elektron. So it's like the next part is problem. We're asked to find actual velocity and I'm going to venture a guess that this lawsuit is going to be comparable to the speed of light. So using its place back a bit, using just a simple connect energy one half and b squared. It's all for V to the route two times k over him. You can play guitar value for Kay and the Mass with electron, and we will get a speed of one point eight seven times ten to the eight muse for seconds. So comparing that to the speed of light, which is three times ten to the eight meters per seconds. You can see this is a very considerable fraction of the speed of light. As I suspected. Let's get an actual gamma factor for it. It's the one point eight seven hey squared over the three each of the eight squared. These are both in meters per second. We'LL get a result in gamma factor of point point two six five Recall that that now recall the momento on the road with relativistic momentum of a particle is gamma envy weaken. They all are results Combine. Um we lose the mast, the electron and the velocity one point eight seven even a second. And this would give us the results of two point one one either native twenty two kilogram meters per second. And if you're wondering, why is this why is this momentum so small? Because we're dealing with a single electron. So the electron mass and as that is complicated nine point E to the negative thirty one kilograms. So even if the velocity is enormous, the mass is so small that if this momentum is going to be pretty small, But when we actually compare this to the non relativistic mo mentum, just envy. We get a result, ng a result we get well, we can simply just suppose, just by the definition of relativistic momentum versus normal momentum, you can see that the ratio is going to be gamma, which is going to be one point two six five. So that's everything there has to find in order to produce a ninety degree scattering of a photonic in a cup, or at least when it's hitting an electron in a conference scattering situation. You know you're going to have a resulting electron velocity that's comparable to the speed of light. That's a significant fraction of it. That's a somewhat interesting result they

Yeah. So in this question we're gonna review how to calculate the momentum of a body given its mass and speed and how to calculate the kind of energy of the body given its mass and speed. So in this question it's given that there is a car With its mass given as 1200 kg and Its speed given as 72 mph. So I'm going to write the mass of the car as m with a subscript C and the speed of the car as we with a subscript C. And I'm not going to use the numbers that are given in this problem because this problem can be solved without using the numbers. So this is the information about the car, There is a mass M. C. And speed we see. Now there is an information given about the Suv. Uh that its mass is which I'm gonna write em with a subscript s and it is given that the mass of the suv is 1.5 times, which is three by two times the mass of the car. And its speed which I'm gonna write as we with a subscript S. Is given as to third the speed of the car. So once again the mass of the car is M. C. The speed of the car as we see the mass of the Suv is M. S. Which is given as three x 2 times the mass of the car. And the speed of the SUv Is v. S. which has given us to 3rd the speed of the car. So to answer the first part, ah the momentum of the SUV which I'm going to write as B with a subscript S. Is will be given as mass of the SUv times the speed of the SUV and mass of the Suv. I'm going to write as three divided by two times the mass of the car. So um subscript s I'm replacing us three x 2. The mass of the car and velocity or the speed of the Suv. I'm going to write us to toll the speed of the car. That is to 3rd the speed of the car. And so if you calculate this these two cancels with this three cancels with this. So this momentum of the Suv actually comes out to be EMC times we see. Whereas the momentum of the car itself would be just the mass of the car times the speed of the car. EMC times we see. So to calculate the ratio uh of the momentum of this ue to the momentum of the car would just be momentum of the issue will be calculated to be M. C. Times we see and the momentum of the car to be EMC times we see. So both these quantities cancel each other and this ratio Will come out to be one. So that stan said to the first part the ratio of the momentum of the suv To the momentum of the car is one. And to answer the second part, once again the kinetic energy of the SUV. And when the writers gave it a subscript S candidate and as half M re squared. Since this is for the Suv this will be M. S. Times we s squared. And once again I'm going to replace M. S. N. V. S. In terms of mm CNBC the mass of the SUV is three x 2 times mass of the car I'm gonna replace M. S. Bye three divided by two. The mass of the car. And the velocity of the SUv. Or the speed of the issue is to third uh the speed of the car. So we as I'm replacing by 2/3 the speed of the car. And this quantity remember that is getting squared. So this was about this uh if you go ahead and calculate this, you should get something like this, leaving the half unchanged. And I'm gonna write M. C. And D. C. Square together. Mhm. So this will be M. C. We c squared. And this quantity three by two mm times To buy three square which is four divided by nine. This will be four divided by now. So if you do this this gives you three and this gives you two. So K. S. The kind of technology of the SUV is actually half EMC we c squared and there is an additional factor of two or three. And the kinetic energy of the car, it's obviously half EMC which is mass off the car times we see square, which is the speed of the car squared. And so once again, if you take the ratio, the kind of take energy of the suv Ks divided by the kinetic energy of the car K C. This will be this number, which I'm going to write half EMC We c squared but with an additional factor of two or three and the kinetic energy of the car is obviously half EMC We c squared. This whole term councils are this whole time and we get the ratio as two over three. So this is the answer to your second part. The kinetic energy of the issue, we divided by the kinetic energy of the car is to over three. So in this question we reviewed how to calculate the momentum of a body and the kinetic energy of the body. I solved this question without using the numbers for masses and speeds. You could do a calculation, you could just use the numbers but I find this to be easier

The related. Sycophantic energies given by K is equal to M C. Squared, divided by square root off one minus B squared, divided by C square minus M. C square. We can shrink this problem. Study this equation into gamma minus one M C square by writing again my simple toe one over one minor square root off three squared, divided by C Square but a off the problem using a known religious stick known only lady Mystic Um, we can write a frantic energy for normal acoustic case that he's the one over two m We square. Or we can say that classical, the classical equation for kind of energy that is only equal to one over and we square. We substitute the values for mass and we 1.67 times 10 to the power minus 27. He's cagey. Is the muscle proton times its speed, which is eight times into the power. Seven meter per second. This gives us the normal is to stick can take energy to be 5.2344 times 10 to the power minus 12 kg meters, where for second square man, then for relativistic case using above equation the road Previously we can Right before electrolytic case, we can write 1.67 times Dental about minus 27 times three times 10 to the power Pete squared, divided by square root off one minus eight times Dental power seven. Divided by squared. Divided by C Square, which is three times 10 to power it square minus UH, 1.67 times down to about minus 27 times three. Poor three times 10 to the power age square. So we find this. We get the chaotic energy relatives to can take. Energy to be 54 and 45164 story times down to the ball or minus 12 kg meter square for a second square if we take the ratio. Did the ratio for really to mystic ionic energy to the normal letters to kinetic energy? There we see five warrant for 64 seven times down to the ball minus 12 divided by 5.344 times dental power minus 12. We get the ratio of 1.0 six, then, ah, for speed. That is a for the speed of a problem for a party that is really easy call to 2.85 times 10 to the power A two meter, which is different than the previous problem. The speed if we change the speed, Uh then the non logistic, antic energy doing the same procedure we get here to be Ah six warrant 78 times 10 to the power minus 11 K g meter square, poor second square. Similarly. Ah, the logistic Arctic energy. For this speed we get here is a 3.3 3.3 times of 10 to the power minus 10. 10. Sorry, not 11 10 kg meter square or second square which if we take the rush off this these values for let you stick to kind Take energy. No logistic. We get here this racial to be 4.8 it which is higher than the biggest problem. Um, we we understand from these results we shows that the ratio for let you stick to normally to security energy just as the higher values in a higher speed when we approach the speed of light but in low speeds the ratio is a too small that we can ignore it when we use ah Newton's law directly. So the summaries for higher speeds, the ratio is greater. Reiter. And for lower speech, the ratio is lower and therefore we can use their known relate to see kinetic energy when we're dealing with our new don't laws.


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