When a proton is traveling at 0.80c,what is the ratio of its kinetic energy to its rest energy?
A)0.60
B)0.67
C)0.80
D)1.2
E)1.7

Question

Quizplus · Accepted Answer

The kinetic energy of a proton traveling at 0.80c is given by:$$K = (\gamma - 1)mc^2$$where $\gamma$ is the Lorentz factor, $m$ is the rest mass of the proton, and $c$ is the speed of light.The Lorentz factor is given by:$$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$$where $v$ is the velocity of the proton.Substituting $\gamma$ and $v = 0.80c$ into the equation for kinetic energy, we get:$$K = \left(\frac{1}{\sqrt{1 - (0.80c)^2/c^2}} - 1\right)mc^2$$$$K = \left(\frac{1}{\sqrt{1 - 0.64}} - 1\right)mc^2$$$$K = \left(\frac{1}{\sqrt{0.36}} - 1\right)mc^2$$$$K = \left(\frac{1}{0.6} - 1\right)mc^2$$$$K = \left(\frac{5}{3} - 1\right)mc^2$$$$K = \frac{2}{3}mc^2$$The rest energy of the proton is given by:$$E_0 = mc^2$$Therefore, the ratio of the kinetic energy to the rest energy is:$$\frac{K}{E_0} = \frac{\frac{2}{3}mc^2}{mc^2} = \frac{2}{3} = 0.67$$Therefore, the correct answer is B) 0.67.

When a Proton Is Traveling at 0