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Protons are positively charged particles found in the nucleus of an atom. In this exercise, the proton is set into motion under the influence of an electric field. When a proton is placed in an electric field, it experiences a force along the direction of the field. This is due to the interaction between the proton's positive charge and the electric field's force lines.
The motion is straightforward: the proton starts from rest and accelerates towards the opposite charge, in this case, towards the negatively charged plate. This type of motion is predictable and is a great example of how charged particles interact with electric fields to yield observable behavior.

Newton's laws of motion are fundamental to understanding how forces affect motion. Here, Newton's second law is particularly relevant. It states that the force acting on an object is equal to the mass of the object multiplied by its acceleration, given by the formula:

• \( F = ma \)

For the proton, we know that the force acting on it is due to the electric field. Therefore, we express this force in terms of the electric field (\( E \)) and the proton's charge (\( q \)):

• \( F = qE \)

Combining these facts gives us a way to relate the electric field to the proton's acceleration through the equation \( E = \frac{ma}{q} \). This relationship is pivotal in calculating how quickly the proton will speed up in response to the electric field.

Kinematics is the study of motion without considering the forces that cause the motion. To find how far and how fast the proton travels between the plates, we employ kinematics equations.

The formula for the distance \( d \) that an object covers when starting from rest under constant acceleration is:

• \( d = \frac{1}{2} a t^2 \)

This equation allows us to compute the acceleration \( a \) when we have values for \( d \) and the time \( t \). Furthermore, once the acceleration is determined, the final speed \( v \) of the proton can be calculated using:

• \( v = at \)

Both these equations illustrate how motion parameters are connected and how kinematics plays a critical role in understanding particle motion under uniform acceleration.

A uniform electric field is one where the electric force is consistent throughout. This means that a charged particle, such as a proton, will experience the same force magnitude and direction regardless of where it is in the field.

In the scenario provided, the uniform electric field exists between two plane parallel plates, with one positively charged and the other negatively charged. When released from rest at the positively charged plate, the proton moves towards the negatively charged plate under the influence of this electric field. The field provides constant acceleration, creating a steady increase in the proton's speed as it traverses the space between the plates.

The magnitude of the uniform electric field can be established by using the relationship between force, charge, and electric field (\( E = \frac{F}{q} \)). This enables us to quantify the field's influence on the proton, showcasing how a uniform electric field can precisely control the motion of charged particles.