The Energy of a Speedy Proton (A) Find the rest energy of a proton in units of e

Question

The Energy of a Speedy Proton (A) Find the rest energy of a proton in units of electron volts. (B) If the total energy of a proton is nine times its rest energy, what is the speed of the proton? (C) Determine the kinetic energy of the proton in units of electron volts. (D) what is the proton's momentum? SOLVE IT (A) Find the rest energy of a proton in units of electron volts. Conceptualize Even if the proton is not moving, it has energy associated with its mass. If it moves, the proton possesses more energy, with the total energy being the sum of its rest energy and its kinetic energy. Categorize The phrase "rest energy" suggests we must take a relativistic rather than a classical approach to this problem. Analyze Use E_R = mc^2 to find the rest energy: E_R = m_p c^2 = (1.6726 times 10^-27 kg)(2.998 times 10^8 m/s)^2 = (1.503 times 10^-10 J) (1.00 eV/1.602 times 10^-19 J) = 938 MeV (B) If the total energy of a proton is nine times its rest energy, what is the speed of the proton? Use E = gamma mc^2 to relate the total energy of the proton to the rest energy: E = 9m c^2 = m c^2/squareroot 1 - u^2/c^2 rightarrow 9 = 1/squareroot 1 - u^2/c^2 Solve for u: 1 - u^2/c^2 = 1/81 rightarrow u^2/c^2 = 80/81 u = squareroot 80/9 c = 0.994 c = 290000000 m/s (C) Determine the kinetic energy of the proton in units of electron volts. Find the kinetic energy of the proton: K = E - m_p c^2 = 9m_p c^2 - m_p c^2 = 8m_p c^2 = 8(938 MeV) = 7504 MeV (D) What is the proton's momentum? Use the following equation to calculate the momentum: E^2 = p^2c^2 + (m_p c^2)^2 = (9m_p c^2)^2 p^2 c^2 = 81(m_p c^2)^2 - (m_p c^2)^2 = 80(m_p c^2)^2 p = squareroot 80 m_p c^2/c = squareroot 80 938 MeV/c = 8389.727 MeV/c Finalize The unit of momentum in part (D) is written MeV/c, which is a common unit in particle physics. For comparison, you might want to solve this example using classical equations. The kinetic energy of the proton is decreased to 1.51 GeV. (a) What is the speed of the proton (in units of c)? 31622.7766 What is the energy of the proton? How does the energy relate to the speed of the proton? c (b) What is its momentum (in units of MeV/c)? MeV/c

Explanation / Answer

a )

given

E = 1.51 GeV = 1.51 X 109 eV

E = 2.419 X 10-10 J

m = 1.67 X 10-27 kg

E = 1/2 mV2

2.419 X 10-10 = 0.5 X 1.67 X 10-27 X V2

V2 = 2.897 X 1017

V = 5.382 X 108 m/sec

the conversion of m/sec into interms of c is

V = 5.382 X 108/ 3 X 108

V = 1.794 C

b )

momentum P = m V

P = 1.67 X 10-27 X 1.794 c

P = 2.99 X 10-27 C

P = 2.99 X 10-27 C / 1.63 X 10-13

P = 1.86 X 10-14 MeV / C

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