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Coulomb's law is fundamental in understanding the forces between charged particles. It describes the interaction force between two charges, stating that the force is directly proportional to the product of the magnitudes of the two charges and inversely proportional to the square of the distance between them. The equation is given as:\[ F = k \frac{q_1 q_2}{r^2} \]where:

• \(F\) is the force between the charges,
• \(k\) is Coulomb’s constant,
• \(q_1\) and \(q_2\) are the amounts of the two charges,
• \(r\) is the distance between the centers of the two charges.

In the context of a proton being repelled by a uranium nucleus, the equation of force changes slightly, as given in the problem statement \( F = \alpha/x^2 \). Here, \(\alpha\) is a proportionality constant that reflects the charge interaction unique to the proton-uranium system. Understanding this force is crucial, as it determines how the proton slows down and changes direction when it approaches the uranium nucleus. This interaction shapes the potential energy landscape as well, altering the proton's journey.

Kinetic energy is the energy possessed by an object due to its motion. The formula for kinetic energy (\( KE \)), particularly relevant for particles such as protons, is:\[ KE = \frac{1}{2} mv^2 \]where:

• \(m\) is the mass of the object,
• \(v\) is the velocity of the object.

In our scenario, the proton starts off with an initial speed of \(3.00 \times 10^5\) m/s, which gives it a certain amount of kinetic energy. As it approaches the uranium nucleus, the kinetic energy changes due to the repelling electrostatic force described by Coulomb's law.
This repulsive force acts opposite to the motion of the proton, decreasing its kinetic energy as potential energy increases. At the closest point to the nucleus, all kinetic energy is temporarily converted to potential energy, meaning the proton's speed becomes zero before it reaccelerates in the opposite direction.

Potential energy in physics refers to the stored energy of position possessed by an object. For charged particles interacting through an electrostatic force, potential energy (\( U \)) can be expressed as:\[ U = -\frac{\alpha}{x} \]In the context of this problem:

• \( \alpha \) is a constant characteristic of the charge interaction,
• \( x \) is the distance between the proton and the uranium nucleus.

As the proton moves closer to the uranium nucleus, it experiences changes in potential energy. Initially, at a distance of 5.00 m, the potential energy is relatively low. However, as it approaches within \(8.00 \times 10^{-10}\) m, the repulsion increases significantly, and so does the potential energy.
The principle of conservation of energy is crucial here, as it states that the total energy (kinetic plus potential) remains constant if only conservative forces are acting. Thus, any loss of kinetic energy as the proton slows due to the force is compensated by an equivalent gain in potential energy.

The interaction between a proton and a uranium nucleus is guided by electrostatic forces, primarily the repulsive force due to their like charges. The characteristics of this interaction can be examined through energy principles and force laws discussed earlier.
The path a proton takes upon approaching a fixed uranium nucleus highlights key physical concepts:

• Initially, the proton has a high speed and high kinetic energy.
• As it moves closer, it experiences a growing repulsive force, as modeled by the specific force equation \( F = \alpha/x^2 \).
• This causes the proton to decelerate, and its kinetic energy is transformed into potential energy.
• Upon reaching a minimum distance, it comes momentarily to rest before being repelled.

The measurement of this closest approach, or how near the proton gets to the nucleus before turning around, is determined when its kinetic energy reaches zero, signifying maximum conversion to potential energy. Eventually, the proton travels back and reaches the original distance with its speed restored, an illustration of energy conservation in its full circle.