When a charged particle like an electron moves through an electric field (**gravity-free space**) with a uniform horizontal velocity, it is subject to a force given by the equation:  

$$\vec F = q \vec E$$
where q is the charge carried by the electron or particle and E is the Electric Field Intensity.  

Here, Electric Field Intensity is inversely proportional to the square of the distance from electron to the side applying the force.

$$\vec E = \frac{kQ}{d^2}$$

As a result, the electron tends to follow a parabolic path in the direction of the force applied as seen below.  

(In this case, the electron has no initial vertical velocity and no horizontal acceleration)

[![Motion of charged particle through an electric field with uniform horizontal velocity](https://i.stack.imgur.com/otsvQ.gif)](https://i.stack.imgur.com/otsvQ.gif)

Such a situation can also be seen in horizontal projectile motion under the influence of gravity where a ball is thrown horizontally from a height, where the force acting on the body is it's own weight or acceleration is due to gravity.

[![Horizontal Projectile Motion](https://i.stack.imgur.com/GqeX2.png)](https://i.stack.imgur.com/GqeX2.png)

However, in this case, the force applied to the body is uniform throughout the motion of the projectile as acceleration due to gravity remains constant close to the surface of the Earth.

My question is as the electron moves in the direction of the force applied, doesn't the distance from the electron to the side applying the force continuously change throughout the motion of the particle and as a result isn't the electric field intensity continuously changing as well with the relation given above?  

So, doesn't it affect the force as well such that it changes at every point of motion of electron unlike in the case of horizontal projectile motion where force due to gravity is always constant?

However, I have seen that the acceleration of electron in the vertical direction is considered to be constant with the relation:
$$\vec a\_{y} = \frac {\vec Eq}{m}$$  

rather than it being variable. I would like an explanation as to why it is constant and not variable like I believe it to be.  

(Source for formulas: [MOTION OF A CHARGED PARTICLE IN ELECTRIC FIELD - A Detailed Analysis](https://www.researchgate.net/publication/346035451_MOTION_OF_A_CHARGED_PARTICLE_IN_ELECTRIC_FIELD_-_A_Detailed_Analysis))