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# An Examination of the Motion of a Particle Through Three Possible Paths

This didnt get a very good grade but it works to help set it up Cycloids Brachistochrones and other Archaic Words Introduction 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 Background 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 x0 y0 and end at xwf ywf These are 06 and 60 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 know that there is no forces in the x direction but we do know that we have to deal with the force of gravity in the y direction Since we have a force it means we have acceleration because of Newtons laws Once we have acceleration we can find the velocity and we can find length independently of the acceleration Once we have velocity and the length of the path we can find out how much time it will take to travel the paths Hypothesis It is well known that a straight line is the shortest path that cam be traveled between

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