Make a graph of the kinetic energy versus momentum for (a) a p...

Question

Make a graph of the kinetic energy versus momentum for (a) a particle of nonzero mass, and (b) a particle with zero mass.

Accepted Answer

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, alpha particles have a charge plus two E and a rest mass of 6.645 multiplied by 10 to the power of negative 27 kg. An accelerator changes the energy of an alpha particle so that its overall energy is 2.5 capital E subscript zero where capital E subscript zero is the rest energy. I calculate the particle's kinetic energy. I I find the momentum magnitude only of the particle II I determine the speed of the alpha particle. Awesome. So we have three separate answers that we're trying to solve for. So our end goal is to find the the particle kinetic energy, the momentum of the particle and the speed of the alpha particle. OK. So we're given some multiple choice answers for III I or for II I and II I and for all the answers for I, they're all in the units of jewels for I I, all the answers are in kilograms multiplied by meters per second. And for II I, all the units are in C or Coolum. Ok. So let's read off our answers to see what our final answer set might be. A is 1.50 multiplied by 10 to the power of negative 94.57 multiplied by 10 to the power of negative and 0.917. B is 8.97 multiplied by 10 to the power of negative 10, 4.57 multiplied by 10 to the power of negative 18 and 0.917 C C is 1.50 multiplied by 10 to the power of negative 92.99 multiplied by 10 to the power of negative 18 and 0.938. And finally, for D is 8.97 multiplied by 10 to the power of negative 10, 2.99 multiplied by 10 to the power of negative 18 and 0.938. OK. So first off, we need to recall the equations for the total energy and for the energy momentum relationship. So the total energy, let's call it equation one is states that E is equal to MC squared plus K where E is the energy K is the kinetic energy M is the mass of the particle and C is the speed of light. So for equation two, we also need to state the energy momentum relationship which states that the energy squared is equal to the mass of the particle multiplied by the speed of light squared, all squared plus the momentum P multiplied by the speed of light all squared. OK. So to solve for, I let us substitute E equals 2. E zero equals 2. MC squared into the total energy equation. And solve for K the kinetic energy. So we need to rearrange the total energy equation to give us K equals E minus MC squared. So when we plug in our known variables, we should get that K equals two point five MC squared minus M C squared. So now we can plug in our numerical values. So let's do that. So K equals 2.5 multiplied by the mass of a particle which is given to us in the problem as 6.645 multiplied by 10 to the power of negative 27 kg multiplied by the speed of light, which is 3.0 multiplied by 10 to the power of eight meters per second, all squared minus M which is 6. five multiplied by 10 to the power of negative kilograms multiplied by the speed of light squared. So 3.0 multiplied by 10 to the power of m per second squared. OK. So when we plug in that long line of numbers into a calculator, we should get 8. multiplied by 10 to the power of negative jewels. So this is our answer for II, I mean for I, so our answer for I is 8.97 multiplied by 10 to the power of negative 10 Jews. OK. So to solve for I I, we need to substitute E equals 2.5 E zero equals 2.5 M C squared into the energy momentum equation relationship. I should say the energy momentum relationship equation and rearrange it to solve for P momentum. So let's do that. So let's plug in our value for E first. So 2.5 mc squared squared equals M C squared, all squared plus P for momentum multiplied by the speed of light C squared. OK. So once we rearrange the Solfur P, we get and when we rearrange and so for P, we get that P equals 5. multiplied by M multiplied by C. So when we plug in our numerical values, we get the momentum is equal to 5.25 or I should say momentum equals the square root of 5.25 multiplied by our mass of the particle, which is given to us at 6.645 multiplied by 10 to the power of negative 27 kg multiplied by the speed of light, which is 3.0 multiplied by 10 to the power of 8 m per second. Which when we plug that into a calculator, we should get the momentum is equal to 4.57 multiplied by 10 to the power of negative 18 kilograms multiplied by meters per second. And that is our final answer for I I, OK. So now we can solve for II I, which is our last step. So II I, so how we solve for II I, we need to recall and rearrange the relativistic energy equation to solve for V. So the relativistic energy equation states that E is equal to MC squared divided by the square root of one minus V squared divided by C squared. So let's rearrange it to solve RV. So let's start to do that. So let's start to simplify. So when we start to simplify, we get that using algebra, we get M multiplied by C squared divided by E equals the square root of one minus V squared divided by C squared. So we can simplify by substituting E equals 2.5 mc squared and by squaring both sides. So when we plug in E equals 2. multiplied by C squared, we get one divided by 6.25 is equal to one minus V squared divided by C square. So when we get isolate V by itself, we get that V is equal to the square root of one minus one divided by 6.25. So and then we multiply it over it multiplied by C. So V equals the square root of one minus one divided by 6. multiplied by C. So when you plug that into a calculator, we should get that V equals 0.917 C. And that is our final answer for II I, all right, we did it. OK. So, so that means looking at our multiple choice answers, our final answer has to be the letter B I equals 8.97 multiplied by 10 to the power of negative 10 jules. I I is 4.57 multiplied by 10 to the power of negative 18 kg multiplied by meters per second. And II I is 0.917 C. So, wow, that was a lot of work, but we did it. Hopefully that helped and I can't wait to see you in the next video. Thanks for watching. Bye.

Make a graph of the kinetic energy versus momentum for (a) a particle of nonzero mass, and (b) a particle with zero mass.

Key Concepts

Kinetic Energy

Momentum

Massless Particles

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