Question 39.9: The Energy of a Speedy Proton (A) Find the rest energy of a ......

(A) 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 Equation 39.24 to find the rest energy:

(B) Use Equation 39.26

E=\frac{m c^2}{\sqrt{1-\frac{u^2}{c^2}}}=\gamma m c^2      (39.26)

to relate the total energy of the proton to the rest energy:

E=3 m_{p} c^{2}=\frac{m_{p} c^{2}}{\sqrt{1-\frac{u^{2}}{c^{2}}}} \rightarrow 3=\frac{1}{\sqrt{1-\frac{u^{2}}{c^{2}}}}

Solve for u :

\begin{aligned}& 1-\frac{u^{2}}{c^{2}}=\frac{1}{9} \rightarrow \frac{u^{2}}{c^{2}}=\frac{8}{9} \\& u=\frac{\sqrt{8}}{3} c=0.943 c=2.83 \times 10^{8} \mathrm{~m} / \mathrm{s}\end{aligned}

(C) Use Equation 39.25 to find the kinetic energy of the proton:

\begin{aligned}K & =E-m_{p} c^{2}=3 m_{p} c^{2}-m_{p} c^{2}=2 m_{p} c^{2} \\& =2(938  \mathrm{MeV})=1.88 \times 10^{3}  \mathrm{MeV}\end{aligned}

(D) Use Equation 39.27 to calculate the momentum:

Finalize The unit of momentum in part (D) is written \mathrm{MeV} / c, which is a common unit in particle physics. For comparison, you might want to solve this example using classical equations.