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Coulomb's Law is a fundamental principle in electrostatics that describes the force between two charged particles. It states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of their separation distance. The mathematical expression for Coulomb's Law is given by:\[ F = \frac{k \cdot |q_1 q_2|}{r^2} \]where:

• \( F \) represents the force between the charges.
• \( k \) is Coulomb's constant \( 8.988 \times 10^9 \, \mathrm{N} \cdot \mathrm{m}^2/\mathrm{C}^2 \).
• \( q_1 \) and \( q_2 \) are the magnitudes of the charges, in this case, both are proton charges \( 1.6 \times 10^{-19} \, \mathrm{C} \).
• \( r \) is the distance between the charges, initially \( 2.50 \, \mathrm{mm} = 2.50 \times 10^{-3} \, \mathrm{m} \).

This law helps us calculate the force that one proton experiences due to the presence of another proton in close proximity, crucial for predicting their behavior under electrostatic interactions.

The electric force is the influence experienced by a charged particle due to the presence of other charges around it. In the context of our exercise, it is the force that acts on the moving proton when it is released. According to the exercise, this force arises because of the repulsion between the two positively charged protons.Electrostatic repulsion occurs because like charges repel. Thus, when the released proton feels a push away from the fixed proton, this is manifested as an electric force.We can determine the magnitude of this electric force using the equation derived from Coulomb's Law:\[ F = \frac{(8.988 \times 10^9) \cdot (1.6 \times 10^{-19})^2}{(2.50 \times 10^{-3})^2} \]Solving this equation, we find the exact force exerted on the released proton, quantifying the electrostatic interaction between them.

Protons, being positively charged particles, experience motion when acted upon by electric forces. In our scenario, the released proton moves because of the electrostatic force created by the fixed proton. Initially, the proton is at rest and is then set into motion by this force. The behavior of this proton's motion can be analyzed using - the concepts of displacement and velocity - changes in acceleration over time. Due to the initial conditions (the proton starting from rest), it accelerates away from the fixed proton. The velocity of the proton increases as the force causes it to move, illustrating a fundamental aspect of electrostatics where charged particles influence each other's motion.

An acceleration-time graph provides an insightful representation of how an object's acceleration changes over time.For the released proton, the initial acceleration can be calculated using the force found from Coulomb's law and Newton's second law:\[ a = \frac{F}{m} \]where \( m \) is the mass of the proton \( 1.67 \times 10^{-27} \, \mathrm{kg} \). The graph initially shows this calculated acceleration.As time progresses, the protons move apart due to repulsive force, which reduces the magnitude of the electrostatic force. Consequently, this results in a decreasing acceleration over time. Thus, the graph would illustrate a declining line after the initial high point of acceleration, showing that as the force decreases, so does the proton's acceleration accordingly.

Newton's Second Law of Motion states that the acceleration of an object is directly proportional to the net force acting on it, and inversely proportional to its mass. This can be expressed using the equation:\[ F = ma \]where:

• \( F \) is the net force acting on the object.
• \( m \) is the mass of the object.
• \( a \) is the acceleration.

Applied to the proton scenario, this law helps us find the initial acceleration of the released proton. By rearranging the formula to solve for acceleration (\[ a = \frac{F}{m} \]) we can substitute the calculated electric force and the proton's mass to determine how fast the proton initially accelerates.This principle not only helps predict the initial behavior of the proton but is essential for understanding how forces affect motion in physics.