1. The problem statement, all variables and given/known data A proton has a kinetic energy of 1000 MeV. In which of the following ranges does its speed v lie? a) v < 0.80c b) 0.80c ≤ v < 0.85c c) 0.85c ≤ v < 0.90c d) 0.90c ≤ v < 0.95c e) v ≥ 0.95c 2. Relevant equations K = (γ-1)mc^2 E0 = mc^2 E^2 = K + E0 3. The attempt at a solution I tried to solve this problem using just the kinetic energy equation, K = (γ-1)mc^2, and solving for the velocity, v in the gamma factor. I used 1000 MeV as the given kinetic energy of the proton and 938.3 MeV/c^2 for the mass of the proton. Doing this, I ended up with a value of v = 0.346c. However, the answer is actually C, which my professor explained with the work below: E = mc^2 + K = 938.3 + 1000 = 1938.3 MeV pc = √(E^2 + E0^2) = 1696 MeV v = (pc^2) / E = 0.875c. I understand his work; everything seems to sense, but my question is this: why is the rest energy added to the kinetic energy of the proton when calculating the velocity? I'm a little confused as to why my method and his method yield a different result, and it seems like that question plays a part in it. Thank you in advance for your help!!