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The concept of electric force lies at the heart of understanding numerous phenomena in physics, especially when studying charged particles. The electric force between two charges is described mathematically by Coulomb's law, which is formulated as \(F = k \cdot \frac{q_1 \cdot q_2}{{d^2}}}\), wherein \(k\) is Coulomb's constant (\(8.99 \times 10^{9} \text{N} \cdot (\text{m}^2/\text{C}^2)\)), \(q_1\) and \(q_2\) represent the magnitudes of the charges involved, and \(d\) is the distance separating them. For protons, which are positively charged particles with a charge of \(1.60 \times 10^{-19} \text{C}\), the electric force is repulsive and will decrease as the distance between them increases.

When working with electric forces in practice, it is important not only to compute the magnitude of the force using Coulomb's law but also to consider its direction, which is along the line connecting the two charges. This understanding is essential when predicting the motion of charged particles in fields.

Acceleration is a measure of how quickly an object's velocity changes. When a proton is released from rest, as in the exercise, it accelerates due to the electric force acting upon it. Here, we can calculate the initial acceleration of a proton using Newton's second law, \( F = m \cdot a \) and Coulomb's law combined.

This involves computing the force between the protons with Coulomb's law and then rearranging Newton's second law to \( a = \frac{F}{m} \) to find acceleration. The mass of a proton (\(m = 1.67 \times 10^{-27} \text{kg}\)) is a constant, meaning the acceleration of the proton can be found by dividing the electric force that we have previously found by this mass. The resulting figure represents the rate at which the proton's velocity will change per second as a result of the electrical interaction with another proton.

When representing motion graphically, an acceleration-time graph offers insights into how an object's acceleration changes over time. In the case of a proton released in the vicinity of another fixed proton, we can imagine how its acceleration evolves. Initially, as the distance between the protons is maximal, the acceleration is minimal. But as the proton moves closer, the force—and consequently the acceleration—increases.

Therefore, the acceleration-time graph for this scenario would start with a lower value at \(t = 0\) and would curve upwards, indicating an increase in acceleration. The curve rises steeper as the distance between the protons reduces, reflecting the inverse square nature of Coulomb's law, where force, and hence acceleration, increases dramatically as distance decreases.

Understanding Velocity Change

A velocity-time graph helps us understand how an object's speed and direction of travel change over time. For a proton accelerating due to electric force, its velocity starts at zero and increases as time progresses. The curve of the graph depends on the acceleration: a constant acceleration produces a straight line, while varying acceleration produces a curve.

Given that the proton's acceleration increases with time due to a decrease in distance from the fixed proton, the slope of the velocity-time graph also increases. In our case, the graph should represent a curve starting at the origin and becoming steeper, indicating that not only is the proton speeding up, but it is doing so at an increasing rate.

One of the foundational principles in physics, Newton's second law, relates the net force acting on an object to its mass and the acceleration it experiences. Stated as \( F = m \cdot a \), this law provides the framework for predicting the motion of objects. In the context of this exercise, we use Newton's second law to determine the proton's acceleration.

It implies that for any object with a constant mass, such as a proton, the net force and acceleration are directly proportional. As a result, if we know the force on the proton due to another charge, we can determine how quickly it will accelerate. This law is fundamental for interpreting interaction outcomes between particles and the resulting motion, making it a cornerstone for exercises involving forces and movements of charged particles.