Explaining Pythagoras' Theorem & Its Impact on Acceleration

[D.A.Peel]

Can anyone explain to me why Pythagoras' Theorem governs the rate of change, of mass, length and time within accelertated bodies?
It's a simple theorem learned by most children by the age of eleven, so one would expect the answer to this question to be quite simple as well.

 

[Ich]

It's actually not the well-known theorem of Euclidean space, but a different version belonging to Minkowski space, where the length of a vector (e.g. a time interval) is calculated from its components using
s² = t² - x² - y² - z². (or -t² + x² + y² + z² as a matter of convention)
In one spatial dimension, this becomes
s² ("true" elapsed time) = t² - x² = t²*(1-v²) (less than elapsed coordinate time).
The same logic gives relativistic mass: it is the "time component" of a vector (Energy-Momentum vector) which has a length equal to the rest mass. Modern usage is to call the time component energy, not relativistic mass.
It's a different situation for length contraction: what we define as "length" is actually not a component of a vector, but a one-dimensional slice of a two-dimensional entity, the measuring rod, which extends both in space and in time. Therefore the different result.

 

[Entropia]

Pythagoras' Theorem is a fundamental mathematical principle that states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem has a wide range of applications in various fields, including physics and engineering.

In terms of acceleration, Pythagoras' Theorem is used to calculate the magnitude of the net acceleration of an object moving in two or three dimensions. In other words, it allows us to determine the total acceleration of an object when it is experiencing both horizontal and vertical accelerations simultaneously.

To understand the impact of Pythagoras' Theorem on acceleration, we need to first understand the concept of acceleration. Acceleration is the rate of change of velocity, which itself is the rate of change of displacement. In other words, acceleration is the measure of how much an object's velocity changes over time. And since velocity is a vector quantity, meaning it has both magnitude and direction, acceleration is also a vector quantity.

Now, let's consider a simple scenario where an object is moving in a straight line with a constant acceleration. In this case, the acceleration is acting only in one direction, and we can easily calculate its magnitude using basic kinematic equations. However, in real-world situations, objects rarely move in a straight line, and their motion is influenced by multiple forces acting in different directions. This is where Pythagoras' Theorem comes into play.

By using Pythagoras' Theorem, we can break down the net acceleration of an object into its horizontal and vertical components. This allows us to better understand the motion of the object and predict its trajectory. In other words, Pythagoras' Theorem helps us to determine the overall impact of multiple forces on an object's acceleration.

Furthermore, Pythagoras' Theorem also has a significant impact on the study of motion and dynamics in general. It serves as the basis for more complex mathematical concepts such as vector addition and decomposition, which are essential in understanding the behavior of accelerated bodies.

In conclusion, Pythagoras' Theorem is a fundamental principle that governs not only the geometry of right triangles but also has a significant impact on the study of acceleration and dynamics. It allows us to break down complex acceleration vectors into simpler components and provides a solid foundation for more advanced mathematical concepts. So, it is no surprise that this simple theorem has such a profound impact on the understanding of