Exponents, logarithms and roots each utilise different notation, which can be confusing for students as it’s almost like learning three different ‘languages’, for no real good reason. The video I am sharing with you, by 3blue1brown, attempts to simplify this notation by using one elegant system, and also explains how maths could be made more accessible by developing cleaner notation.
A Brachistochrone Curve is the curve that would carry a bead from rest along the curve, without friction, under constant gravity, to an end point in the shortest amount of time.
The Brachistochrone problem was one of the earliest problems posed in calculus of variations. The solution, which is a segment of a cycloid, was found individually by Leibniz, L’Hospital, Newton and both the Bernoulli’s.
The following solutions are taken from wikipedia.
Johann Bernoulli’s Solution
Johann Bernoulli used Fermat’s principle that “the actual path between two points taken by a beam of light is the one which is traversed in the least time” in order to derive the brachistochrone curve. He did this by considering the path that a beam of light would take in a medium where the speed of light increases due to a constant vertical acceleration equal to g.
Due to the conservation of energy, , where y is the vertical distance. Furthermore, the law of refraction gives us a constant (vm) of the motion for a beam of light in a medium of variable density:
Rearranging this gives us
which can be manipulated to give
If we assume that the beam, with coordinates (x,y) departs from the origin and reaches a maximum speed after falling a vertical distance D:
we can rearrange the equation to give us the following:
which is the differential equation of an inverted cycloid generated by a circle of diameter D, as required.
Jakob Bernoulli’s Solution
Jakob Bernoulli’s approach was to use second order differentials to find the condition for the least time. The differential triangle formed by the displacement along the path, the horizontal displacement and the vertical displacement is a right-handed triangle, therefore:
Differentiating this gives
Consider the follow diagram:
The horizontal separation between paths along the central line is d2x.
The diagram gives us two separate equations:
For the path of the least time, these times are equal hence their difference is equal to zero.
Today I wanted to share a quick video following on from my blog post about illusions. It is about a US magician and inventor Mark Setteducati, “whose toys and puzzles have earned him patents and accolades, personal expression and a mastery of maths have been key”.
I was thinking of perhaps writing a post about mathematical magic; let me know if you’d be interested in that!
Today’s post was a short one but this week I have my final exams (finally!), so hopefully I will be able to focus more on my blog. M x
As a big advocate of the beauty of mathematics, I though I would share this video that I found by the Paris-based design studio Parachutes. It highlights the mathematics hidden in our everyday lives, from the patterns of storm clouds to the binary code used in our computers.