Speed of proton with kinetic energy 1000 MeV

[mrdoodle]

Homework Statement

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

Homework Equations

K = (γ-1)mc^2
E0 = mc^2
E^2 = K + E0

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!

 

[TSny]

Hi, mrdoodle. Your method will work. You must have just made an error in the manipulations.

Your professor's method is based on the relationship between total energy and momentum. Total energy, E, is defined as rest energy plus kinetic energy.

 

[mrdoodle]

Ah, sorry I should've checked my work more thoroughly. Turns out I forgot to subtract 1 from the gamma factor even though I had the correct equation. Thank you for the explanation!