An electron is accelerated to a speed where its total energy is 219 times larger
ID: 1949992 • Letter: A
Question
An electron is accelerated to a speed where its total energy is 219 times larger than its rest energy.How many times is the kinetic energy of the electron larger than its rest energy?
218 You are correct.
What is the kinetic energy of the electron?
Hint: Try to use MeV units, don't convert to Joules. The rest energy of the electron is 0.511 MeV.
What is the speed of the electron in terms of the speed of light?
(In this part of the problem you have to enter your answer with a precision of seven or eight digits!)
What is the momentum of the electron measured in MeV/c units? (Enter only a number without the MeV/c unit.)
Explanation / Answer
Lets use the following formulas, remember that GAMMA = 1/sqrt(1-v^2/c^2) Ill go a little out of order and solve for the speed first Rest Energy = mc^2 Total Energy = GAMMA * mc^2 We are told that the total energy is 219 times the rest energy so lets write the equation for that 219*(Rest Energy) = Total Energy 219*(mc^2) = GAMMA * mc^2 We can cancel the mc^2 from both sides to get: GAMMA = 219 1/sqrt(1-v^2/c^2) = 219 Multiply both sides by sqrt(1-v^2/c^2) to get: 1 = 219 sqrt(1-v^2/c^2) Now divide both sides by 219 to get: 1/219 = sqrt(1-v^2/c^2) Now square both sides to cancel the square root on the right (1/219)^2 = 1-v^2/c^2 1/47961 = 1-v^2/c^2 Now subtract 1 from both sides to get: 1/47961 -1 = -v^2/c^2 -47960/47961 = -v^2/c^2 Now multiply both sides by -1 to get: 47960/47961 = v^2/c^2 Now take the square root of both sides to get: sqrt(47960/47961) = v/c 0.9999895748 = v/c Multiply both sides by c The velocity of the electron is v = 0.9999895748c =299996872.4 m/s Now lets solve for the kinetic energy of the electron using the following formula: K = (GAMMA -1)* mc^2 We already know from earlier on that GAMMA = 219 and we also know that mc^2 equals the rest energy of the electron which is .511 MeV so lets plug these values in K = (219 - 1) * (.511 MeV) K = 111.398 MeV Now lets solve for the momentum of the electron using the following formula for relativistic momentum: P = GAMMA*m*v We know that GAMMA is 219, mass is the rest mass, and v is the velocity which we solved to be 0.9999895748c We are told that the rest energy is 0.511 MeV, so lets use that to solve for the rest mass E = mc^2 0.511 MeV = mc^2 0.511 MeV/c^2 = m Now lets plug in this rest mass and our velocity and GAMMA to solve for momentum P = GAMMA * m * v P = 219 * (0.511 MeV/c^2) * 0.9999895748c P = 111.9078333 MeV/c So just to recap: 1) The kinetic energy is 218 times greater than the rest energy 2) The kinetic energy of the electron equals 111.398 MeV 3) The speed of the electron in terms of the speed of light is 0.9999895748c 4) The momentum of the electron is 111.9078333 MeV/c Hope this was helpful!
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