Compute the kinetic energy of a proton (mass 1.67 * 10-27 kg) using both the nonrelativistic and relativistic expressions, and compute the ratio of the two results (relativistic divided by nonrelativistic) for speeds of (a) 8.00 * 107 m/s and (b) 2.85 * 108 m/s.

Question

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. So relativistic effects will be experienced when an object exceeds 0.1 C where C is the speed of light, the mass of a neutron is 1.6749, multiplied by 10 to the power of negative 27 kg, determine the ratio K subscript N R divided by K subscript R R relativistic and N R non relativistic when the neutron moves at I 0.2 C and I I 0.8 C. OK. Awesome. So that's our end goal to determine the ratio K subscript N R divided by K subscript R. OK. So we're given some multiple choice answers here and they're all given as answers for I and for answers for I, I. So let's read them off to see what our final answer might be. Our final answers. I should say A is 1.3 and 2.8. B is 24.3 and 0. C is 0.412 and 1.33. And D is 0.971 and 0.480. Awesome. So, first off, we need to recall a few equations, we need to recall the equations for non relativistic kinetic energy, relativistic kinetic energy and the Lourens factor or it's also known as the relativistic factor equation. So let's start writing those down. We'll denote it as kinetic energy as capital K. So the kinetic energy for non relativistic E conditions is equal to one half multiplied by the mass multiplied by the velocity square. And then the kinetic energy for relativistic conditions K subscript li little R lower case R is equal to dam up minus one multiplied by the mass multiplied by the speed of light squared. And finally, the relativistic factor equation states that gamma equals one divided by the square root of one minus the velocity squared divided by the speed of light squared. OK. So let's make a quick note here that C the numerical value for C is 3.0 multiplied by 10 to the power of eight m per second. And that our mass of the neutron, which I'm gonna denote it as little m lowercase M subscript capital N equals 1. multiplied by 10 to the negative power of kg. Awesome. OK. So now we could start solving for I so to do that let's start off by solving for gamma using the relativistic factory equation. And let's plug in all of our known variables into that and solve for gamma. So gamma equals one divided by one minus which note that for I, the velocity is 0.2 C. So let's do that 0.2 C squared, 0.2 C squared divided by C squared note that the, the speed alike the CS cancel out. So when you plug that into a calculator, you should get 1.206. Awesome. So now we can sol for the ratio for K subscript N R divided by K subscript R. So K and R divided by R. So now we can sol for that ratio by plugging in all of our known variables. OK. So let's do that shall we? So K and R kinetic energy for non relativistic E E conditions and kinetic energy for relativistic conditions equals OK. So for non relativistic for kinetic engine, so the non relativistic Connecticut energy, let's use our equation. Here is one half multiplied by the mass of the neutron which was 1.6749, multiplied by 10 to the power of negative kilograms multiplied by our velocity which is 0. multiplied by C which is 3.0 multiplied by 10 to the power of eight m per second. And that is all squared. OK. Divided by gamma which the value for gamma, we determined it to be 1. minus one. And this is the relativistic kinetic energy equation multiplied by one point which is the mass of the neutron, 1.6749, multiplied by 10 to the power of minus kg multiplied by the speed of light squared, 3.0 multiplied by 10 to the eighth power meters per second squared. Awesome. So when you plug that into a calculator, your final answer should be for the ratio, it should equal 0.971. So that is our answer for I. So to get I I our second answer, we need to repeat the same process as a as we just did using our new values which in this case for I I, the velocity is 0.8 C. So let's do that. So let's start by solving for gamma using the relativistic factor equation. So it's one divided by the square root of one minus our velocity, which was 0.8 C squared divided by C squared. So the CS cancel out the speed of light cancels out. And when you plug that into a calculator, you should get 1.6667. Awesome. So now let's solve for our ratio for the non relativistic kinetic energy divided by the relativistic kinetic energy. So let's do that. So it's one half multiplied by the mass of the neutron, 1.6749 multiplied by 10 to the power of negative 27 kg, multiplied by 0.8, multiplied by the speed of light 3.0, multiplied by 10 to the eighth power meters per second. All squared divided by our gamma value which we found out was 1. minus one multiplied by the mass of the neutron. 1.6749 multiplied by 10 to the minus 27 kg multiplied by the speed of light squared, which is 3.0 multiplied by 10 to the eighth power meters per second squared. Awesome. So when you plug up all of that into a calculator, the ratio should equal for non relativistic kinetic energy divided by relativistic kinetic energy. Your value should be 0.480. And that is our final answer for I I Awesome, we did it. Ok. Ok. So that means our final answer should be D I is 0.971 and I I is 0.480. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Thanks for watching. Bye.

Compute the kinetic energy of a proton (mass 1.67 * 10-27 kg) using both the nonrelativistic and relativistic expressions, and compute the ratio of the two results (relativistic divided by nonrelativistic) for speeds of (a) 8.00 * 107 m/s and (b) 2.85 * 108 m/s.

Verified Solution

Key Concepts

Kinetic Energy

Relativistic Effects

Ratio of Energies