Cycloids, Brachistochrones, And Other Archaic Words Essay, Research Paper
This didnt get a very good grade but it works to help set it up.
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 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 Newton s 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.
It is well known that a straight line is the shortest path that cam be traveled between two points, but is it the quickest? If we apply a little bit if common sense to the problem we can make a reasonable guess to answer this problem. If you want to get to California from New York and you are going by land, you are going to have to cross the Rocky Mountains in some way. There are two main options, you could either dig a tunnel (which is rather time intensive) thru the mountains, or you could climb over the mountains. The path of least resistance and less time would be to climb over the mountain. Therefore a reasonable hypothesis would be that it is likely that the cycloid and ellipse could take less time than the straight line.
The parametric equations in Table 1 are all standard parametric equations of the paths that we have chosen to examine. There parametric equations go from the starting point (0,6) to the ending point (6,0). When finding the parametric equations for the line segment, we applied the use of basic vector mathematics. To get the equations for the cycloid we had to solve for two variables using initial conditions that were given. In finding the ellipse s parametric equations we had to find the semi-major and semi-minor axis to get the final result of the parametric equations. (The calculations are located in Appendix A).
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
theorem. To find the length of the cycloid and ellipse we
can apply the arc length integral. (Calculations and further
explanation in Appendix B).
Another component we need to find the time is the velocities of particles along each path. Which can be done using the acceleration due to gravity and knowing the distance change in the y direction. The equation gives us velocity which gives us the equations in Table 3. Velocity is still an equation because we don t know what y(w) equals, except in equation form. The calculations of velocity are in the appendix C, included in the calculations for time.
Now that we have the length of the paths and the velocities along the paths we can find the time that it takes for a particle to travel the path. The time integral is: , if you perform
a unit analysis on the equation you will see
that it is dimensionally correct. (The unit
analysis and the other time calculations are in
Appendix C). Table 4 shows the results of using the time integral. The results show that the cycloid and ellipse are both faster traveled than the line segment.
The results listed in Table 4 prove that the cycloid and elliptic paths are faster, than the line segment, which had the shortest length. The results support the hypothesis of the writer, because the cycloid and the elliptical path do take less time than the line segment.
Parametric Equations for line segment:
To find a parameterization of the line from (x(0), y(0)) to (x(wf), y(wf)), we use these two points (0,6) and (6,0) respectively, and find the vector between the two.
to find the parameterization, you use the fact that a standard parameterization of a line through P(xo, yo) and parallel to v = Ai+Bj+Ck is equal to x = xo+At and y = yo+Bt. Which gives us that:
Parametric equations of the cycloid:
(x(0),y(0)) = (6,0)
(x(wf), y(wf)) = (0,6)
Using these equations and MVT s implicit plotter we can get this graph where the y-axis is representing a, and the x-axis represents wf. The blue line is x(w) and the red is y(w) :
From this graph we can predict that the intersection of the two curves is somewhere near a = y = 35 and wf = x = 2.5. With this guess we can go to MVT s Numerical rootfinder and get the exact location of the intersection of the two curves.
The numerical rootfinder tells us that the intersection is at a = 33.722 and wf = 2.412. wf is the ending point that we are going to use. a is the acceleration of the particle along the path of the cycloid. Since a is a constant, we can put it into the equations to simplify the equations, which gives:
Parametric Equations for Ellipse:
The semi-major axis is represented by c and the semi-minor axis is represented by b in the ellipse equations listed above.
With the information given we can solve the equations for b and c.
Since the set of equations that we are dealing with are parametric equations, we can say that the c in the x(t) and the y(t) are the same number. We can apply the same thing to the b in both of the equations, the only difference in the solving for c and b is that we have to use x(90o) = 2.
Which gives us the equations:
Length of the line segment:
The length of the line segment is found by applying the Pythagorean theorem.
Length of the cycloid:
The length of the cycloid and the ellipse is found by applying the arc length integral:
A numerical integrator can be used to gain the final answer to the arc length integral.
Length of the ellipse:
The arc length integral can also be used to find the length of the curve formed by the ellipse.
To find the time that it takes a particle to travel along the paths we first need to find the velocity which is obtained from the equation: . Since the velocity in the x-direction does not change and we know that due to gravity the velocity in the y-direction changes we can use that fact to our advantage. The equation represents the velocity between two points due to gravity. After we have found the velocity we can apply the time integral. . If you do a unit analysis on this integral it makes sense, m stands for meters, and s stands for seconds. The first derivative of a function is the velocity of that function.
Time of the line segment:
Time of cycloid:
Time of ellipse: