An electron has a kinetic energy that is twice its rest energy. Determine its speed.
A) 0.76 c
B) 0.81 c
C) 0.94 c
D) 0.54 c
E) 0.87 c

Question

Quizplus · Accepted Answer

According to Einstein's Theory of Relativity, the total energy (E) of a particle with rest mass (m) is given by the equation: E^2 = (mc^2)^2 + (pc)^2, where c is the speed of light and p is the momentum of the particle. Since the electron has kinetic energy (K) equal to twice its rest energy (mc^2), we can write:
E = K + mc^2 = 2mc^2 + mc^2 = 3mc^2
We can rearrange the first equation as: p = sqrt(E^2 - (mc^2)^2)/c. Substituting the expression for E and simplifying, we get:
p = sqrt(8m^2c^4)/c = 2mc
The speed (v) of the electron is given by: v = p/m. Substituting the expression for p and m, we get:
v = 2mc/m = 2c
Therefore, the speed of the electron is 2 times the speed of light, which violates the laws of physics. However, we can use the formula for relativistic kinetic energy to find the actual speed (v'):
K = (gamma - 1)mc^2, where gamma is the Lorentz factor: gamma = 1/sqrt(1 - v'^2/c^2)
Substituting K = 2mc^2 and solving for v', we get:
v' = c*sqrt(1 - 1/gamma^2) = c*sqrt(1 - 4/9) = c*sqrt(5/9) = 0.943c
Therefore, the answer is C: 0.94 c.

An Electron Has a Kinetic Energy That Is Twice Its