If we take a closer look at this derivation we can see that there are many assumptions made. If we avoid these assumptions we can derive a different set of equations using boundaries to define the model that we are using. The application of boundaries and using separation of variables allows us to attain an equation of velocity as a function of time. Not assuming the indefinite integral also rids any confusion that a constant of integration brings into play. While this equation is not far off from the kinetic energy equation, we have managed to turn the velocity equation into a function of time.
Velocity is the first time derivative of position so the derivation becomes a second integration.
Once again we have obtained an equation as a function of time. The increasing velocity gives this graph a parabolic curve upward as the electron position changes with respect to time. From our first derivation, we now have two equations that give velocities as a function of time and position. The use of the proper electric field equation for the boundaries given is also an important factor in determining the proper velocity solutions. There are many types of scenarios in which an electric field can be created.