Analysis: The Enigma Of The Fastest Path

The Enigma of the Fastest Path
“Mathematics directs the flow of the universe, lurks behind its shapes and curves, holds the reins of everything from tiny atoms to the biggest stars.”
― Edward Frenkel, Love and Math: The Heart of Hidden Reality I would often wonder what would it be like to travelling at the speed of light. It could possibly only be a mere aspiration because I realized that achieving that speed was close to impossible. Definitely not a job worthy of a 17-year-old. It is no espionage that humans always try to determine the shortest path, given that our psyche works that way. The completion of a task with minimal effort and greater quality. It is then when I pondered if I could displace myself equivalent to any other path and
I spent months trying to envisage the analogy I made and soon forgot about it because I thought that this problem was too complex to understand and wasn’t significant enough. I live in the desert country of Qatar where the probability of finding a falcon is significantly high. It was one day when I noticed a Falcon land to catch its prey, mice that was seemingly trying to find its way to its burrow. This happened within an instant, but for the keen observer that I am, I managed to point out a detail corresponding to the manner in which the bird landed. It followed a certain path, a certain curve. I had assumed that the fastest path was linear given that it was the path with the least distance. So why didn’t this bird follow a straight-line path? Would it not have reached its prey in a shorter duration of time given that it would have to cover lesser distance that way? I was inquisitive. I wanted to know more about this phenomenon. It is then when I sought enlightenment from my Physics and Mathematics teachers. They told me that there existed a path, which favored speed and would thus, enable the
The respective length and breadth of the triangle are dx and dy. Henceforth, on using the Pythagoras theorem:

As the above formulated equation cannot be converted in the form of y=f(x), I will define this in terms of a parametric equation. Equations (1) and (2) can provide us with the subsequent length of any curve with the function f(x) or in the form of parametric equations x(t) and y(t). Now, in the case of a straight line that passes through the two aforementioned co-ordinates (0,0) and (π,2). Using equation (1): In this case, ‘h’ is a function of x in terms of t (u=x(t)). In order to calculate the time required, an equation for ‘v’ (velocity) needs to be formulated. Given that velocity refers to the rate of change of displacement, or s’(t)
I need to use the Fundamental Theorem of Calculus in the third equation to derive the equation for the velocity. The Fundamental Theorem states:

Hence, (Using chain rule and substituting ‘u’ with x(t))
As it has been conjectured that the only force acting on the ball is gravity. From energy conservation in