Given information:
- A particle of mass m and charge q is released from rest at the origin.
- The particle travels along the z-axis.
- The distance travelled by the particle along the z-axis is d.

Explanation:
To find the speed of the particle when it has travelled a distance d along the z-axis, we can use the principles of work and energy. The work done on the particle is equal to the change in its kinetic energy.

Work done:
The work done on the particle can be calculated using the formula:
Work = Force * Distance

In this case, the force acting on the particle is the electric force. The electric force experienced by a charged particle in an electric field is given by the formula:
Force = q * Electric field

Since the particle is moving along the z-axis, the electric field is in the same direction. Therefore, the electric field is constant and given by:
Electric field = V / d

where V is the potential difference between the origin and the point where the particle has travelled a distance d.

Substituting the values, the force can be expressed as:
Force = q * (V / d)

Therefore, the work done on the particle is:
Work = q * (V / d) * d
Work = q * V

Kinetic energy:
The change in kinetic energy of the particle is given by:
Change in kinetic energy = Final kinetic energy - Initial kinetic energy

Since the particle is released from rest, its initial kinetic energy is zero. Therefore, the change in kinetic energy is equal to the final kinetic energy.

Final kinetic energy:
The final kinetic energy can be calculated using the formula:
Final kinetic energy = (1/2) * m * v^2

where v is the final velocity of the particle.

Equating work done and change in kinetic energy:
Since the work done on the particle is equal to the change in its kinetic energy, we can equate the two expressions:

q * V = (1/2) * m * v^2

Solving for v, we get:

v = sqrt(2 * (q * V) / m)

Final Answer:
The speed of the particle when it has traveled a distance d along the z-axis is given by:

v = sqrt(2 * (q * V) / m)