Anderson Video - Accelerating a Proton Relativistically

Professor Anderson
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<font color="#ffffff">Hi guys, welcome back. Let's take a look at one of your homework problems that</font> <font color="#ffffff">relates to the energy problem that we just talked about, and this is number</font> <font color="#ffffff">2629 and it says how much work is required to accelerate a proton from rest, up to a</font> <font color="#ffffff">speed of 0.984C.</font> <font color="#ffffff">Okay, so they give us the following, the initial speed is zero,</font> <font color="#ffffff">the final speed is 0.984C,</font> <font color="#ffffff">and they want to know how much work is required to do that.</font> <font color="#ffffff">So let's think about a proton, here's a proton sitting there</font> <font color="#ffffff">and now we're gonna start pushing on it and eventually it's going to be moving along</font> <font color="#ffffff">at speed V final. When it is sitting there initially, how much energy is in</font> <font color="#ffffff">the system? Well, it's not moving so there's no momentum there but there is</font> <font color="#ffffff">of course the rest energy of the proton.</font> <font color="#ffffff">When it starts moving it now has a final energy</font> <font color="#ffffff">which is gamma MC squared. </font> <font color="#ffffff">Okay, so this gamma thing looks really critical to</font> <font color="#ffffff">understanding what the total energy is, and if we can calculate gamma, we can</font> <font color="#ffffff">probably figure out how much work is required because the work is just going</font> <font color="#ffffff">to be the change in energy.</font> <font color="#ffffff">We know what the final is, its gamma MC squared. We know what the initial was,</font> <font color="#ffffff">it's just MC squared, and so we get gamma minus 1 times MC squared.</font> <font color="#ffffff">So let's figure out what gamma is.</font> <font color="#ffffff">Gamma is 1 over the square root 1 minus V squared over C squared,</font> <font color="#ffffff">and if we are moving at 0.984 C, what do we get for our gamma?</font> <font color="#ffffff">We have a 0.984 C, we're going to divide by C and square that stuff, therefore the</font> <font color="#ffffff">C's will cancel out, we get 1 over the square root 1 minus of 0.984 squared, and why</font> <font color="#ffffff">don't you guys punch that into your calculator and tell me what you get.</font> <font color="#ffffff">Anybody get an answer for that one?</font> <font color="#ffffff">>> (student speaking) 5.6.</font> <font color="#ffffff">>> 5.6. Okay, somebody else confirm that, 5.6?</font> <font color="#ffffff">Okay, so work</font> <font color="#ffffff">we said was gamma minus 1 MC squared,</font> <font color="#ffffff">and so it's 4.6 times M times C squared,</font> <font color="#ffffff">but what is M? M is the mass of a proton.</font> <font color="#ffffff">It's the rest mass of a proton and if we flip open the inside cover of your book</font> <font color="#ffffff">you can figure out what the rest mass of a proton is, it's 1.67</font> <font color="#ffffff">times 10 to the minus 27 kilograms.</font> <font color="#ffffff">We're going to multiply by C squared</font> <font color="#ffffff">three times 10 to the 8th, we're all in SI units here and so we can punch this into</font> <font color="#ffffff">our calculator and see what we get.</font> <font color="#ffffff">>> (student speaking) I have a question.</font> <font color="#ffffff">>> Yeah?</font> <font color="#ffffff">>> (student speaking) Where did you get the 4.6?</font> <font color="#ffffff">>> 5.6 minus one, yeah.</font> <font color="#ffffff">And we got to square the C, right? Yeah.</font> <font color="#ffffff">And if somebody gets a number, shout it out.</font> <font color="#ffffff">>> (student speaking) 6.9 times 10 to the minus 10.</font> <font color="#ffffff">>> 6.9 times ten to the minus ten</font> <font color="#ffffff">and we're in SI units so that should be joules, and that's what I get</font> <font color="#ffffff">here on Wolfram Alpha so let's try that, let's punch it in and see if that's what we get.</font> <font color="#ffffff">6.9 times ten to the minus ten --</font> <font color="#ffffff">good, sounds correct. Okay, that</font> <font color="#ffffff">doesn't sound like a lot of energy but remember a proton is really really</font> <font color="#ffffff">pretty small, right? It's ten to the minus 27 kilograms, so that's a pretty</font> <font color="#ffffff">significant amount of energy for moving that thing, that fast. Okay, Part B says,</font> <font color="#ffffff">what would be the momentum of this proton? All right, that's not too bad.</font> <font color="#ffffff">All right, momentum is what? Well, momentum is gamma mV.</font> <font color="#ffffff">Okay, so in this case it's going to be gamma MV final and for our numbers we</font> <font color="#ffffff">already know what gamma is. Gamma was 5.6,</font> <font color="#ffffff">mass of the proton 1.67 times ten to the minus 27 kilograms,</font> <font color="#ffffff">V final is right there, 0.984 C,</font> <font color="#ffffff">which is three times 10 to the eighth.</font> <font color="#ffffff">All right, punch all those numbers into your calculator and let's see what we get.</font> <font color="#ffffff">And I got 2.76 times 10 to the minus 18. Anybody else get that one? Okay.</font> <font color="#ffffff">What are the units of momentum, remember its kilogram meters per second, right?</font> <font color="#ffffff">Gamma is unit-less, mass is kilograms, V is of course meters per second.</font> <font color="#ffffff">So let's try that one and see if we're right.</font> <font color="#ffffff">And I think it needs a dot in there</font> <font color="#ffffff">and that says we're right, okay? So try that</font> <font color="#ffffff">with your numbers, you will of course have a different value for your V-final,</font> <font color="#ffffff">which means you will have a different value for your gamma.</font> <font color="#ffffff">Questions on that one? Everybody okay?</font> <font color="#ffffff">Let's ask you a follow-up question, how much momentum would it have classically?</font> <font color="#ffffff">That one's easy, right? Classically P is equal to mV</font> <font color="#ffffff">and so it's the mass of the proton, 1.67 times ten to the minus 27</font> <font color="#ffffff">times this V, 0.984 times C, three times ten to the eighth.</font> <font color="#ffffff">That's going to be a number that is</font> <font color="#ffffff">significantly less than the one that we just calculated. Right, and you know</font> <font color="#ffffff">exactly how much less, if I just take away the gamma and do that calculation,</font> <font color="#ffffff">what do I get?</font> <font color="#ffffff">I get what I had before, 2.76 times ten to the minus 18 divided by the</font> <font color="#ffffff">gamma that we calculated for ours, which was 5.6.</font> <font color="#ffffff">Okay, so relativistic momentum takes significantly more oomph to get these</font> <font color="#ffffff">things going, significantly more energy to get these</font> <font color="#ffffff">particles going and it's because of that gamma factor that it keeps increasing.</font>
<font color="#ffffff">Hi guys, welcome back. Let's take a look at one of your homework problems that</font> <font color="#ffffff">relates to the energy problem that we just talked about, and this is number</font> <font color="#ffffff">2629 and it says how much work is required to accelerate a proton from rest, up to a</font> <font color="#ffffff">speed of 0.984C.</font> <font color="#ffffff">Okay, so they give us the following, the initial speed is zero,</font> <font color="#ffffff">the final speed is 0.984C,</font> <font color="#ffffff">and they want to know how much work is required to do that.</font> <font color="#ffffff">So let's think about a proton, here's a proton sitting there</font> <font color="#ffffff">and now we're gonna start pushing on it and eventually it's going to be moving along</font> <font color="#ffffff">at speed V final. When it is sitting there initially, how much energy is in</font> <font color="#ffffff">the system? Well, it's not moving so there's no momentum there but there is</font> <font color="#ffffff">of course the rest energy of the proton.</font> <font color="#ffffff">When it starts moving it now has a final energy</font> <font color="#ffffff">which is gamma MC squared. </font> <font color="#ffffff">Okay, so this gamma thing looks really critical to</font> <font color="#ffffff">understanding what the total energy is, and if we can calculate gamma, we can</font> <font color="#ffffff">probably figure out how much work is required because the work is just going</font> <font color="#ffffff">to be the change in energy.</font> <font color="#ffffff">We know what the final is, its gamma MC squared. We know what the initial was,</font> <font color="#ffffff">it's just MC squared, and so we get gamma minus 1 times MC squared.</font> <font color="#ffffff">So let's figure out what gamma is.</font> <font color="#ffffff">Gamma is 1 over the square root 1 minus V squared over C squared,</font> <font color="#ffffff">and if we are moving at 0.984 C, what do we get for our gamma?</font> <font color="#ffffff">We have a 0.984 C, we're going to divide by C and square that stuff, therefore the</font> <font color="#ffffff">C's will cancel out, we get 1 over the square root 1 minus of 0.984 squared, and why</font> <font color="#ffffff">don't you guys punch that into your calculator and tell me what you get.</font> <font color="#ffffff">Anybody get an answer for that one?</font> <font color="#ffffff">>> (student speaking) 5.6.</font> <font color="#ffffff">>> 5.6. Okay, somebody else confirm that, 5.6?</font> <font color="#ffffff">Okay, so work</font> <font color="#ffffff">we said was gamma minus 1 MC squared,</font> <font color="#ffffff">and so it's 4.6 times M times C squared,</font> <font color="#ffffff">but what is M? M is the mass of a proton.</font> <font color="#ffffff">It's the rest mass of a proton and if we flip open the inside cover of your book</font> <font color="#ffffff">you can figure out what the rest mass of a proton is, it's 1.67</font> <font color="#ffffff">times 10 to the minus 27 kilograms.</font> <font color="#ffffff">We're going to multiply by C squared</font> <font color="#ffffff">three times 10 to the 8th, we're all in SI units here and so we can punch this into</font> <font color="#ffffff">our calculator and see what we get.</font> <font color="#ffffff">>> (student speaking) I have a question.</font> <font color="#ffffff">>> Yeah?</font> <font color="#ffffff">>> (student speaking) Where did you get the 4.6?</font> <font color="#ffffff">>> 5.6 minus one, yeah.</font> <font color="#ffffff">And we got to square the C, right? Yeah.</font> <font color="#ffffff">And if somebody gets a number, shout it out.</font> <font color="#ffffff">>> (student speaking) 6.9 times 10 to the minus 10.</font> <font color="#ffffff">>> 6.9 times ten to the minus ten</font> <font color="#ffffff">and we're in SI units so that should be joules, and that's what I get</font> <font color="#ffffff">here on Wolfram Alpha so let's try that, let's punch it in and see if that's what we get.</font> <font color="#ffffff">6.9 times ten to the minus ten --</font> <font color="#ffffff">good, sounds correct. Okay, that</font> <font color="#ffffff">doesn't sound like a lot of energy but remember a proton is really really</font> <font color="#ffffff">pretty small, right? It's ten to the minus 27 kilograms, so that's a pretty</font> <font color="#ffffff">significant amount of energy for moving that thing, that fast. Okay, Part B says,</font> <font color="#ffffff">what would be the momentum of this proton? All right, that's not too bad.</font> <font color="#ffffff">All right, momentum is what? Well, momentum is gamma mV.</font> <font color="#ffffff">Okay, so in this case it's going to be gamma MV final and for our numbers we</font> <font color="#ffffff">already know what gamma is. Gamma was 5.6,</font> <font color="#ffffff">mass of the proton 1.67 times ten to the minus 27 kilograms,</font> <font color="#ffffff">V final is right there, 0.984 C,</font> <font color="#ffffff">which is three times 10 to the eighth.</font> <font color="#ffffff">All right, punch all those numbers into your calculator and let's see what we get.</font> <font color="#ffffff">And I got 2.76 times 10 to the minus 18. Anybody else get that one? Okay.</font> <font color="#ffffff">What are the units of momentum, remember its kilogram meters per second, right?</font> <font color="#ffffff">Gamma is unit-less, mass is kilograms, V is of course meters per second.</font> <font color="#ffffff">So let's try that one and see if we're right.</font> <font color="#ffffff">And I think it needs a dot in there</font> <font color="#ffffff">and that says we're right, okay? So try that</font> <font color="#ffffff">with your numbers, you will of course have a different value for your V-final,</font> <font color="#ffffff">which means you will have a different value for your gamma.</font> <font color="#ffffff">Questions on that one? Everybody okay?</font> <font color="#ffffff">Let's ask you a follow-up question, how much momentum would it have classically?</font> <font color="#ffffff">That one's easy, right? Classically P is equal to mV</font> <font color="#ffffff">and so it's the mass of the proton, 1.67 times ten to the minus 27</font> <font color="#ffffff">times this V, 0.984 times C, three times ten to the eighth.</font> <font color="#ffffff">That's going to be a number that is</font> <font color="#ffffff">significantly less than the one that we just calculated. Right, and you know</font> <font color="#ffffff">exactly how much less, if I just take away the gamma and do that calculation,</font> <font color="#ffffff">what do I get?</font> <font color="#ffffff">I get what I had before, 2.76 times ten to the minus 18 divided by the</font> <font color="#ffffff">gamma that we calculated for ours, which was 5.6.</font> <font color="#ffffff">Okay, so relativistic momentum takes significantly more oomph to get these</font> <font color="#ffffff">things going, significantly more energy to get these</font> <font color="#ffffff">particles going and it's because of that gamma factor that it keeps increasing.</font>