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Find (a) the speed and (b) the momentum of a proton with kinetic energy $500 \mathrm{MeV}$.

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Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

Numerade Educator

McMaster University

starting with the kinetic energy we write down the formula and we have the EMC squared times gamma minus one. So we insert the value for the protons rest energy, EMC squared and we proceed as follows. So we have 500 MTV equal to 938 m E V, which is the rest energy times gamma minus one. So we see the units will cancel and this little simplify as follows. So 500 over 938 will then equal gamma minus one. From here, we we just have to solve for the velocity for the speed. So we add one over and now we just have gamma equaling 500 over 938 plus one. And then we go ahead and we square both suck both sides and let me scroll down and we're continuing. We conflict both sides, and we have one divided by one plus 9 500 over 938 squared. And then we subtract that from one, and we have b squared overseas where equaling that. And finally we have the velocity Equalling the following quantity that you see on the right hand side And when you put this into a calculator, you finally get V equaling 0.758 times the speed of light. So now we can get mo mentum. With this velocity we get, we have gamma times envy, and we substitute in our value for the velocity. And we get 938 envy overseas where which is the mass the rest mass of the proton times 0.758 c divided by the square root of one minus 10.758 squared. And then when we simplify this, we can get the mo mentum in two units to system of units. We can write it as 1000 and 90 m e v oversee, or we can convert this toe s I units. So we multiply 1000 and 90 times 10 to the sixth, and then we multiply that by 1.6 times sending the negative 19 because we want to convert to Jules and then divide by the speed of light. And that's how you would convert that s I units and you finally obtained in kilogram meters per second 5.813 times 10 to the minus 19. And that's the result in S I units

Georgia Institute of Technology - Main Campus