Cycloids, Brachistochrones, and other Archaic Words

Everyone with a decent math background knows that a straight line is the shortest distance between two points. Intuitively, many would think that it would also be the path taken that took the shortest amount of time. We will show that the time that it takes to traverse cycloidal and elliptic path is faster than that of a straight line. We will do this by using the Pythagorean theorem, the arc length integral, basic vector knowledge, and knowledge of velocity-time relationships.

The motion of the particle will be examined by using three different possible paths, a straight-line segment, an elliptic path, and a cycloidal path. Each vary in length, but the all start and end at the same points. They start at (x(0), y(0)) and end at (x(wf), y(wf)). These are (0,6) and (6,0) respectively. It is possible to track the velocity and time of the particle because of the forces due to gravity. In general motion physics we have learned that unless there is a force acting on an object in any direction, then there will be no change in velocity. In the case we are studying we kno

a unit analysis on the equation you will see

To find the time we need to find the lengths of all the paths that are all listed in Table 2. For the line segment we can apply the Pythagorean

A numerical integrator can be used to gain the final answer to the arc length integral.

Some common words found in the essay are:
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Approximate Word count = 1375
Approximate Pages = 6 (250 words per page double spaced)