Introduction
In this problem, we are given a scenario where a moving electron collides with a stationary electron, resulting in the creation of an electron-positron pair. We are asked to use a relativistic calculation to show that the minimum required kinetic energy for this process is 6mc^2, where m is the rest mass of the electron.

Derivation
Let's consider the conservation of momentum and energy in this collision. Initially, we have one moving electron with momentum p1 and kinetic energy KE1, and one stationary electron with zero momentum and zero kinetic energy. After the collision, we have two particles: a moving electron-positron pair with total momentum p2 and total kinetic energy KE2.

Conservation of Momentum
Since momentum is conserved in the collision, we have:
p1 + 0 = p2
p1 = p2

Conservation of Energy
According to the conservation of energy, the initial kinetic energy of the system should be equal to the final kinetic energy. In the initial state, the kinetic energy is given by:
KE1 = (γ1 - 1)mc^2

In the final state, the total kinetic energy of the electron-positron pair is given by:
KE2 = (γ2 - 1)(2mc^2)

Since the particles have the same velocity after the collision, their Lorentz factor (γ) will be the same. Therefore, γ1 = γ2 = γ.

Minimum Kinetic Energy
To find the minimum kinetic energy, we need to minimize KE2 by setting the derivative of KE2 with respect to γ equal to zero.

d(KE2)/dγ = 2mc^2(γ - 1) = 0

Solving this equation, we find γ = 1, which corresponds to the minimum kinetic energy. Therefore, the minimum kinetic energy is given by:

KEmin = (γ - 1)(2mc^2) = (1 - 1)(2mc^2) = 0

However, this result contradicts the assumption that the particles have the same velocity after the collision. Therefore, the minimum kinetic energy cannot be zero.

Correcting the Minimum Kinetic Energy
Let's consider the case where the minimum kinetic energy is nonzero. In this case, γ > 1.

KEmin = (γ - 1)(2mc^2)

Since γ > 1, we can write γ = 1 + ε, where ε is a small positive quantity.

KEmin = (1 + ε - 1)(2mc^2)
= ε(2mc^2)

Since ε is small, we can approximate it as ε ≈ 0.

KEmin ≈ 0(2mc^2) = 0

Therefore, the minimum kinetic energy is zero, which contradicts our assumption. Hence, the minimum kinetic energy cannot be achieved.

Conclusion
In conclusion, we have shown that the minimum kinetic energy required for the process of an electron colliding with a stationary electron and creating an electron-positron pair is zero. This result is obtained by considering the conservation of momentum and energy in the collision and analyzing the minimum kinetic energy through a relativistic calculation.