The equation of motion for a bungee jump from the bridge will depend
on weight of the person and the stretchiness and length of the bungee cord.
The bungee cord is 200 feet long. When tested, it stretched 8 feet when 4 lbs was attached. Let�s test the claim with a 160 lb person. So the cord will stretch to 320 feet with the person attached and just hanging (it will stretch much more than that when the person has a velocity when the cord starts to stretch, but when the person comes to rest it would be at 320 feet).
Royal Gorge Bridge is 955 feet above the Arkansas River.
During freefall there is air resistance as the jumper is typically in a spread eagle position but for simplicity in this problem, we are neglecting air resistance since it is not large. Another assumption is the person is stepping off a platform (so no initial velocity).
At 3.536 seconds, the bungee cord begins to stretch. The equation of motion giving position in feet tseconds after the cord begins to stretch is . So this equation is valid only after the free fall period . (Note that in this equation s(0) = 200 feet which is the position at the end of the freefall period).
The amusement park owner is claiming velocities of 100 mph and a jump coming within 25 feet of the river. Can you substantiate his claims, or could he be accused of false advertising? Back up your answer showing how you used calculus in your determination.
A couple of hints: Find the velocity by hand (it is great practice). If the numbers in the position function make it hard to think of the derivative rules, sub in letters i.e. e?at(Acosbt+Bsinbt) and use the product rule (chain rule will be needed while doing the product rule). Then when you have the derivative you can sub in the numbers. It is ok to use technology for finding the maximum velocity function by looking at the graph or to find the zeros of the velocity function. A great program is https://www.desmos.com/calculator (Links to an external site.) where you can just type in your velocity function and then you can drag the graph or zoom in or out and if you click on a point it will tell you value (so you can click on the maximum or the zero and it will give you the point).
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