A proton (with a rest mass m = 1.67 times 10^-27 kg) has a total energy that is

Question

A proton (with a rest mass m = 1.67 times 10^-27 kg) has a total energy that is 4.00 times its rest energy. Part A What is the kinetic energy E_k of the proton? Express your answer in billions of electron volts to three significant figures. Part B What is the magnitude of the momentum p of the proton? Express your answer in kilogram-meters per second to three significant figures. Part C What is the speed v of the proton? Express your answer as a fraction of the speed of light to four significant figures.

Explanation / Answer

given

  E = 4 mc^2
gama *mc^2 = 4 mc^2
gama = 4

(a)

E_k = [gama - 1]mc^2= [4 - 1](1.67*10^-27)(3*10^8)^2= 4.51*10^-10 J
                              = (4.51*10^-10/1.6*10^-19) eV
                              = 2.82*10^9 eV = 2.82 GeV

(c)

1/Sqrt[1 - (v/c)^2] = 4
                     1 - (v/c)^2 = 1/16
                          (v/c)^2 = 1 - (1/16)
                          (v/c)^2 = 15/16
                                 v/c = 0.97
                                    v = 0.97 c
(b)

P = gama * mv = 4 (1.67*10^-27 Kg) ( 0.97 * 3 * 10^8) = 1.94 * 10^-18 kg m/s

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