Electrons are accelerated through a potential difference of 750 kV, so that their kinetic energy is 7.50 * 105 eV. (a) What is the ratio of the speed v of an electron having this energy to the speed of light, c? (b) What would the speed be if it were computed from the principles of classical mechanics?

Video transcript

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem. A particle accelerator uses a 2.0 mega volt potential difference to accelerate a hydrogen ion proton giving it a kinetic energy of 5.60 multiplied by 10 to the fifth power electron volts. I express the protons velocity magnitude as a fraction of the speed of light C I I use Newtonian laws of motion to determine the protons speed. OK. So we have two goals that we need to accomplish to answer this problem. So we need to express the protons velocity magnitude as a fraction of the speed of light. And we need to use an Newtonian laws of motion to determine the proton speed. Awesome. So when you're given some multiple choice answers here, and all the answers for I are given in units of C and all the answers for I I are given in units of meters per second. So let's read off our multiple choice sensors to see what our final pair of an, like our final answer pair will be. So A is 0.346 and 1. multiplied by 10 to the power seven B is 0.346 and 5.18 multiplied by 10 to the power of six. C is 0.413 and 1.4 multiplied by 10 to the power of seven and D is 0.413 and 5.18 multiplied by 10 to the power of six. OK. So first off, it's important to note that the potential difference in this problem is very large. Thus, it is implied that the relativistic speeds will be responsible for the acceleration of the protons. We also need to recall that the relativistic formula for kinetic energy, which we're gonna call equation one is capital K for kinetic energy is equal to one divided by the square root of one minus the speed of the proton squared divided by the speed of light squared minus one multiplied by the mass of the proton multiplied by the speed of light squared. OK. And let's make a quick note that the numerical value for the speed of light is 3. multiplied by 10 to the power of eight m per second. OK. Awesome. And also we need to make a quick note because it isn't given us, it isn't given to us in the prom itself. But the rest mass of a proto of a proton. So the rest mass of a proton is equal to 1.673 multiplied by 10 to the power of negative 27 kilograms. Awesome. We also need to recall, we'll call it equation to the classic expression for kinetic energy. So the classic expression for kinetic energy is the kinetic energy is equal to one half multiplied by the mass of the proton multiplied by the speed of the proton squared. So first off, let's start off by converting the kinetic energy of proton of the proton to Jews. OK. So let's note that there in one electron volt, there is 1. multiplied by 10 to the power of negative 19. So it's given to us in the pro that the kinetic energy is equal to 5.60 multiplied by 10 to the fifth power electron volts. OK. So we need to use dimensional analysis really quick to convert our kinetic energy which is an electron volts to jewels. So let's do that. So we're trying to convert 5.60 multiplied by 10 to the fifth power electron volts. And we need to multiply that by 1.602 multiplied by 10 to the power of negative 19 Joles. And there is that many Jews in one electron volt. So divided by one electron volt. So 1.602 multiplied by 10 to the power of or negative, I should say 10 to the negative 19th power Jews divided by one electron volt. So the electron bolts cancel out. So that means our kinetic energy in jewels is 8.9712 multiplied by 10 to the power of negative 14 jewels. So so we can finally start to solve for I. So to solve for I, we must use the relativistic formula for the kinetic energy and rearrange that equation to solve for B. So we need to use equation one. So in order to simplify equation one, let's start by adding one to both sides of the equation and then dividing MC squared to both sides. So let's do that. So when we do that, it'd be K divided by MC squared plus one equals one divided by the square root of one minus the speed velocity squared divided by the speed of light squared. OK. So let's start by solving for the kinetic energy divided by the mass of the proton divide uh the mass of the proton multiplied by the speed of light squared plus one. So the numerical value for that, let's solve for that, let's plug in all of our known variables. So we determine the kinetic energy to be 8.9712, multiplied by 10 to the power of negative 14 jewels. And we need to multiply and then divide it by. And then we need to write down the mass of the proton the rest mass which was one point 673, multiplied by 10 to the native 27th power kilograms multiplied by the speed of light squared, which is 3.0, multiplied by 10 to the power of eight meters per second squared. And then don't forget the plus one. So when we plug that into a calculator, we should get 1.6. Awesome. So our equation should look like this. Now, 1.6 equals one divided by the square root of one minus V squared divided by C squared. OK. So using algebra, we can rearrange this equation to isolate V by itself. And when we do that, we should get V is equal to C the speed of light multiplied by the square root of one minus one divided by 1.6 all squared. OK? So when you plug that into a calculator, V should equal 0.346. C means we need to leave it in terms of C OK. So that is our answer for I I hooray, we did it. OK. So now let's start to solve four I I, OK. So to solve for I I, we need to use the equation for the classic kinetic energy. So the classic expression for kinetic energy and we need to rearrange it to solve for V like we did above. So when we do that, when we rearrange it, we should get that V equals two multiplied by K divided by M. So V equals the square root of two multiplied by K divided by M. So that's what we should get when we rearrange the class, the classic expression for kinetic energy to solve from V. So now we could plug in our numerical values to sol for V. So let's do that. So two multiplied by the kinetic energy which was 8.9712, multiplied by 10 to the power of negative divided by the rest mass which was 1.6, 73 multiplied by 10 to the power of negative 27 kilograms. So when you plug that into a calculator, we should get 1. multiplied by 10 to the power of seven m per second. Awesome. So that is the proton speed. All right, we did it. We found the answer for I I. So that means our final answer has to be A I is 0.346 C and I I is 1.4 multiplied by 10 to the power of seven m per second. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.