Engineering >> Mechanical EngineeringSpiral Thrill Rideby Benjamin Overton
Submitted : Fall 2012 This project analyzes the dynamics of a roller coaster initially at rest going down a logarithmic spiral for 720o then coming to a rest at the bottom of the track.The coaster begins at 34 meters high and comes to rest at ground level.Tracking the motion uses a cylindrical coordinate system, two dimensional polar coordinates with an added z-axis for height. I derived equations for the arc length, velocity, and acceleration as a function of both time and θ.Due to the acceleration being constant, once the values of each function were plugged in all of the θ’s canceled out.Integrating this with respect time gave the velocity in terms of time.This is set to equal the velocity function involving θ, and by integrating both sides the variable t can be isolated and solved for.The total time elapsed is then plugged into the velocity equation to give the final velocity when θ=4π.Plotting the velocity squared versus the height revealed a direct connection between the decreases in height to the increases in velocity.The graph decreased linearly with z on the vertical axis and V2 on the horizontal.This demonstrates the law of conservation of energy, as the gravitational potential energy decreased the velocity increased proportionally. Related Links:
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