Explanation:

When a particle of charge Q and mass m travels through a potential difference V from rest, the final momentum of the particle can be calculated using the following equation:

Final momentum (p) = √(2mQV)

This equation is derived from the conservation of energy principle, which states that the total energy of a system remains constant. In this case, the initial energy of the particle is zero (since it is at rest), and the final energy is equal to the potential energy gained by the particle as it moves through the potential difference.

Step-by-step calculation:

1. Calculate the potential energy gained by the particle:

Potential energy (U) = QV

2. Apply the conservation of energy principle:

Initial energy (Ei) = 0

Final energy (Ef) = U = QV

Total energy (E) = Ei + Ef = QV

3. Calculate the final kinetic energy of the particle:

Final kinetic energy (Kf) = E - U = QV - QV = 0

Since the final kinetic energy is zero, the final momentum of the particle is also zero.

4. Apply the conservation of momentum principle:

Initial momentum (pi) = 0

Final momentum (pf) = 0

Since the initial momentum is zero, the final momentum of the particle is also zero.

Conclusion:

The final momentum of a particle of charge Q and mass m traveling through a potential difference V from rest is zero. This is because the particle gains potential energy as it moves through the potential difference, but this energy is converted entirely into potential energy and not kinetic energy. As a result, the particle does not gain any momentum.