Theorem:

Both gravitational and electrostatic forces can be described by a generalized force equation, potential energy equation, and escape velocity equation, by aliasing their respective constants into a unified framework.

Definitions:

1. Gravitational Force: The force between two masses $M$ and $m$, separated by a distance $r$, is given by Newton’s law of gravitation:

   $$F_{\text{gravity}} = \frac{GMm}{r^2}$$

   Where $G$ is the gravitational constant.

2. Electrostatic (Coulomb) Force: The force between two charges $Ze$ and $e$, separated by a distance $r$, is given by Coulomb’s law:

   $$F_{\text{coulomb}} = \frac{kZe^2}{r^2}$$

   Where $k$ is Coulomb’s constant, and $Z$ is the atomic number.

3. Potential Energy: The potential energy associated with each force can be written as:

   • Gravitational potential energy: $$E_{\text{gravity}} = -\frac{GMm}{r}$$
   • Electrostatic potential energy: $$E_{\text{coulomb}} = -\frac{kZe^2}{r}$$

4. Escape Energy: The energy required to overcome the attractive force in each system:

   • Gravitational escape energy is equivalent to the kinetic energy required to escape the gravitational pull: $$E_{\text{escape, gravity}} = \frac{GMm}{r}$$
   • Ionization energy is equivalent to the energy required to free an electron from a nucleus: $$E_{\text{escape, coulomb}} = \frac{kZe^2}{r}$$

5. Escape Velocity:

   • Gravitational escape velocity is derived by equating kinetic energy to the escape energy: $$v_{\text{escape, gravity}} = \sqrt{\frac{2GM}{r}}$$
   • Ionization energy can be interpreted in a similar way, but instead of velocity, we use energy to describe the ionization process.

Generalized Force and Potential Energy:

We introduce a generalized force $F_X$ and potential energy $E_X$, where:

• $A$ is a generalized constant that can represent either $G$ (for gravity) or $k$ (for electrostatic forces).
• $B$ is a generalized product of interacting properties, which could represent $Mm$ (for gravity) or $Ze^2$ (for electrostatic forces).

Generalized Force:

The force can be written as:

Where:

• $A$ represents either $G$ (gravitational constant) or $k$ (Coulomb’s constant).
• $B$ represents either $Mm$ (mass interaction) or $Ze^2$ (charge interaction).

For gravitational systems:

For electrostatic systems:

Thus, the generalized force $F_X$ holds for both gravitational and electrostatic systems.

Generalized Potential Energy:

The potential energy associated with these forces is given by:

For gravitational systems:

For electrostatic systems:

Thus, the generalized potential energy $E_X$ holds for both gravitational and electrostatic systems.

Generalized Escape Energy and Velocity:

Generalized Escape Energy:

The energy required to escape (whether escaping a gravitational field or ionizing an electron) is given by:

For gravitational systems:

For electrostatic systems:

Thus, the generalized escape energy equation holds for both systems.

Generalized Escape Velocity:

In both systems, the escape velocity can be derived by equating kinetic energy to escape energy. The generalized escape velocity $v_X$ is given by:

For gravitational systems:

For electrostatic systems, while we don't usually express ionization energy as a velocity, we can still use the same form for generalization.

Thus, the generalized escape velocity equation holds for both systems.

Generalized Total Energy:

The total energy in both systems can be written as the sum of kinetic energy and potential energy:

For gravitational systems:

For electrostatic systems:

Thus, the generalized total energy equation holds for both systems.

Conclusion:

We have successfully shown that by defining a generalized force, potential energy, escape energy, and escape velocity, we can write a unified set of equations that describe both gravitational and electrostatic interactions. These equations hold regardless of whether the underlying force is gravitational or electrostatic by properly defining the generalized constants $A$ and $B$.

Thus, the following generalized equations work for both gravitational and electrostatic systems:

1. Generalized Force: $$F_X = \frac{A B}{r^2}$$
2. Generalized Potential Energy: $$E_X = -\frac{A B}{r}$$
3. Generalized Escape Energy: $$E_{\text{escape}} = \frac{A B}{r}$$
4. Generalized Escape Velocity: $$v_X = \sqrt{\frac{2A B}{r}}$$
5. Generalized Total Energy: $$E_{\text{total}} = \frac{1}{2}mv^2 - \frac{A B}{r}$$

This proof shows that gravitational and electrostatic forces, despite arising from different physical phenomena, follow a similar mathematical structure and can be unified through generalized equations.