A particle has rest mass 6.64 * 10-27 kg and momentum 2.10 * 10-18 kg•m/s. (a) What is the total energy (kinetic plus rest energy) of the particle? (b) What is the kinetic energy of the particle? (c) What is the ratio of the kinetic energy to the rest energy of the particle?

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 know. In order to solve this problem, a lithium atom has a rest mass of 1.17 multiplied by 10 to the power of negative 26 kg for an atom whose momentum magnitude is 1.15 multiplied by 10 to the power of negative 18 kilograms multiplied by meters per second. Determine I the sum of the kinetic and rest energy of the atom. I I the atoms kinetic energy and I I I, the ratio of the atoms rest energy to the kinetic energy. OK. So we're given some multiple choice answers. Let's read them off to see what our final answer might be. A is I 3.24 multiplied by 10 to the power of minus five. Jes I I is 3.24 multiplied by 10 to the negative five power Jews. And I I I is 3.25 multiplied by 10 to the power of minus five or to the power of negative five. And then B is I 1.15 multiplied by 10 to the power. Negative nine Jews, I, I is 9.20, multiplied by 10 to the power of negative 11 Jews. And I I, I is 0.920. C is uh for I is 1.15 multiplied by 10 to the negative to the 10 to the power of negative nine Jules, I, I is 9.20, multiplied by 10 to the power of negative 11 Jules. And I I I is 11.4. And finally, D is I 3.24 multiplied by 10 to the power of negative five Jews, I, I is 3.24 multiplied by 10 to the power of negative five Jules. And I I I is negative 1.0. OK. So first off, let's recall the equation for the energy momentum relationship. And that states that the energy squared is equal to the rest mass M multiplied by the speed of light squared all to the power of two plus the momentum represented by P multiplied by the speed of light all to the power too. OK. So let's note here that the numerical value for the speed of light is 3.0 multiplied by 10 to the power of eight m per second. OK. And let's also make a quick note that the momentum P in this case, as given to us. And the problem is 1.5 multiplied by 10 to the power of negative 18 kilograms multiplied by meters per second. And that the rest mass is equal to 1.17, multiplied by 10, 1.7 multiplied by 10 to the power of mind to the negative kilograms. Awesome. OK. So we also need to recall the equation for total energy which states that E is equal to M multiplied by C squared plus K. So it says energy is equal to the rest mass multiplied by the speed of light squared plus K where K is the kinetic energy. So now we can finally start solving for I. So we to solve for I, we will use the energy momentum relationship to determine the sum of the kinetic and rest energies of a lithium atom. OK. So first off to get rid of the squared and the energy momentum relationship with square, both sides to get rid of the squared on the E. And when we do that, we should get that E is equal to the square root of M multiplied by C squared, alter the power of two plus P multiplied by C squared. OK. So now we can plug in our, all of our known variables to solve for E. So let's do that. OK. So the rest mass as we determined was 1. multiplied by 10 to the negative 26 power kilograms multiplied by the speed of light. Let's put a bracket multiplied by the speed of light which was 3.0, multiplied by 10 to the eighth power meters per second squared plus the momentum which was 1. multiplied by 10 to the power of negative kilograms multiplied by meters per second, multiplied by the speed of light. OK. So when you plug all of that into a calculator, you should get 1. multiplied by 10 to the power of negative nine Jews. So the sum of the kinetic and rest energies of the atom is equals 1.145 multiplied by 10 to the negative nine power jewels. OK. So to solve for, I, I let's do that. So we need to use the total energy equation to solve for the kinetic energy. So when we rearrange the total energy equation to solve for K, we should get that the kinetic energy equals the energy minus the rest mass multiplied by the speed of light squared. So let's plug in our known variables. So we determined our energy to be 1.145 multiplied by 10 to the power of negative nine Jews. And our rest mass was 1. multiplied by 10 to the power of negative 26 negative kg multiplied by the speed of light, which it's 3.0 multiplied by 10 to the eighth power meters per second squared. So when we plug that into a calculator, we should get 9.20 multiplied by 10 to the negative 11th power jaws which is the Adams kinetic energy. OK. So now we can start solving for I I I our last little piece of information that we need to solve for. So we need to note that the rest energy of the atom is the rest mass multiplied by the speed of light squared. Thus, the ratio of the atom's rest energy to the kinetic energy can be written as such that MC squared, the rest mass multiplied by the speed of light squared divided by the kinetic energy. OK. So let's plug in our known variables which is for the rest mass was 1.17 multiplied by 10 to the power of negative kilograms multiplied by the speed of light squared, which is 3.0 multiplied by 10 to the eighth power meters per second squared. And then we determined that our kinetic energy was 9.20 multiplied by 10 to the power of negative 11 Jules. So when you plug that into a calculator, the ratio of the rest mass multiplied by the speed of light squared, divided by the kinetic energy will equal 11.4. So the ratio equals 11.4. Thus the atoms rest energy to the kinetic energy, that ratio is 11.4. Fantastic. We did it. OK. So that means that our final answer has to be C I is 1.15 multiplied by 10 to the power of negative nine Jules. I I is 9.20 multiplied by 10 to the power of negative 11 jewels and I I I is 11.4. Awesome. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.