A given scenario of a person falling from a height into a pit attached to a rope can be modeled by the second-order non-homogeneous differential equation mh'' + 5h' + 120h = mg where m is mass of person (in pounds), g is gravity (in feet), 5 is air resistance, and 120 is spring constant of rope. Our initial conditions are h(0)=100 and h'(0)=0 for our initial height and velocity at time t=0. The pit is 75 ft deep and ground level represents height of 0. Problem is, people of different masses will reach different heights and velocities, and we want to make sure that bungee jumping will be safe for everyone (say for people between 20 and 300 pounds). We don't want people to hit the bottom of the pit nor fly up past where they initially jumped from. And We don't want them to experience acceleration greater than say 3 g's. Question is, what is a process for finding the parameters that would meet these conditions and ensure the jumpers would be safe? In other words, how would I figure out what parameters to change (air resistance, spring constant, and/or rope length) and what to change them to to meet the safety conditions?
A given scenario of a person falling from a height into a pit attached to a rope can be modeled by the second-order non-homogeneous differential equation mh'' + 5h' + 120h = mg where m is mass of person (in pounds), g is gravity (in feet), 5 is air resistance, and 120 is spring constant of rope. Our initial conditions are h(0)=100 and h'(0)=0 for our initial height and velocity at time t=0. The pit is 75 ft deep and ground level represents height of 0. Problem is, people of different masses will reach different heights and velocities, and we want to make sure that bungee jumping will be safe for everyone (say for people between 20 and 300 pounds). We don't want people to hit the bottom of the pit nor fly up past where they initially jumped from. And We don't want them to experience acceleration greater than say 3 g's. Question is, what is a process for finding the parameters that would meet these conditions and ensure the jumpers would be safe? In other words, how would I figure out what parameters to change (air resistance, spring constant, and/or rope length) and what to change them to to meet the safety conditions?
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