When an electron and a proton are accelerated through a potential difference of 100 kV, the final speeds of the electron and the proton will be different.

To understand why, we can use the equation for the kinetic energy of a particle:

KE = (1/2)mv^2

Where KE is the kinetic energy, m is the mass of the particle, and v is the velocity of the particle.

Since both the electron and the proton start from rest, their initial kinetic energies are zero.

When accelerated through the potential difference, the electron and the proton gain the same amount of kinetic energy. However, the mass of the proton is much larger than the mass of the electron, so the final velocity of the electron (Ve) will be higher than the final velocity of the proton (Vp). This means that Ve > Vp.

Therefore, the correct answer is option 1: Ve > Vp.

It is important to note that we can determine the relation between the final velocities of the electron and the proton based on their masses and their initial potential energies.