Why are Functions instead of equations of variations used to model physical quantities? Explain it in detail along with mathematical expressions and equations.

Answer 1

Functions are typically used instead of equations of variations to model physical quantities because they provide a more flexible and practical way to represent these quantities. Equations of variation, also known as functional equations, involve the study of equations that express relationships between functions. While they can be powerful tools for understanding certain mathematical concepts, they may not always be the most convenient or efficient way to represent physical quantities.

To illustrate this, let's consider an example of modeling the trajectory of a projectile. The motion of the projectile can be described by certain physical laws, such as Newton's laws of motion. In this case, using equations of variation would involve solving a functional equation that expresses the relationship between the position, velocity, and time. This equation may be quite complex and difficult to solve, especially for more intricate motions.

On the other hand, using functions provides a simpler and more intuitive representation of the trajectory. The position of the projectile as a function of time can be written as a mathematical function, such as:

𝑝(𝑡) = 𝑣₀𝑡 + 0.5𝑎𝑡²

where 𝑣₀ represents the initial velocity, 𝑎 represents the acceleration (which depends on the force acting on the projectile), and 𝑡 represents time. This equation is derived from the basic kinematic equation, and it allows us to directly calculate the position of the projectile at any given time.

The advantage of using functions in this case is that they provide a more concrete representation of the physical quantity (position) as a mathematical object that can be easily manipulated and used for further analysis. Functions allow us to analyze the behavior of the physical quantity with respect to different variables, such as time, and to perform various operations, such as differentiation and integration, which are essential in physics.

Furthermore, employing functions enables us to model more complex systems in a straightforward manner. For example, for a projectile launched with a non-constant acceleration (e.g., when considering air resistance), using functions allows us to easily incorporate such variations by defining a more complex function for acceleration. This provides a more accurate representation of the physical reality compared to an equation of variation, which might not be able to capture these nuances as effectively.

In summary, functions are preferred over equations of variations when modeling physical quantities because they provide a more convenient and flexible representation. Functions allow for direct calculations, straightforward manipulation, and analysis of various aspects of the physical quantity, enabling a deeper understanding and more accurate modeling of the underlying phenomena.