Relativistic Energy of an Electron

To calculate the velocity at which the total relativistic energy of an electron becomes 1.25 times the rest energy, we need to use the equation for the total relativistic energy of a particle:

E = γmc²

where E is the total relativistic energy, γ is the Lorentz factor, m is the rest mass of the electron, and c is the speed of light.

Calculating the Lorentz Factor

The Lorentz factor γ can be calculated using the equation:

γ = 1 / √(1 - (v/c)²)

where v is the velocity of the electron.

Equating the Relativistic Energy

We are given that the total relativistic energy is 1.25 times the rest energy, so we can write:

E = 1.25mc²

Substituting the expression for E with the equation for relativistic energy, we get:

1.25mc² = γmc²

Dividing both sides of the equation by mc², we find:

1.25 = γ

Solving for Velocity

Substituting the value of γ into the equation for the Lorentz factor, we have:

1.25 = 1 / √(1 - (v/c)²)

Squaring both sides of the equation, we get:

1.5625 = 1 / (1 - (v/c)²)

Taking the reciprocal of both sides, we have:

1/(1.5625) = 1 - (v/c)²

Simplifying, we find:

0.64 = 1 - (v/c)²

Rearranging the equation, we get:

(v/c)² = 1 - 0.64

(v/c)² = 0.36

Taking the square root of both sides, we find:

v/c = √(0.36)

v/c = 0.6

Calculating the Velocity

To find the velocity of the electron, we multiply both sides of the equation by c:

v = 0.6c

Thus, the velocity at which the total relativistic energy of the electron becomes 1.25 times the rest energy is 0.6 times the speed of light.