Real Problem: Bungee jumping attracts those who like the strong sense of danger as part of their entertainment. Jumps have been made from high bridges, tall cranes, and hot air balloons. During the first part of the jump, jumpers are pulled toward earth by gravity, accelerating in free fall, much like a sky diver. This free fall is related to the quality of the jump: free fall for as long as possible and reach a speed as high as possible. The free fall continues only as long as the bungee cord remains slack, at which time it begins to stretch, applying an upward force that decelerates the jumper. This decelerating force increases as the bungee cord stretches. If all goes well, jumpers are brought to a stop before they hit the ground. At this point, the jumper begins to oscillate up and down until forces acting on him/her sum to zero.
Part 1 - Let L be the length of the bungee cord, d the length of its stretch, k the cord's stiffness, m the mass of the jumper, and g (9.8 m/sec2) the acceleration due to gravity. We assume that there are only two forces acting on the jumper: "gravitational potential energy" due to the jumper's position above the ground, and "elastic potential energy" due to the restoring force of the stretched bungee cord. Further, there is no air resistance (drag). In addition, we assume that the bungee cord behaves like a linear spring. When the cord is stretched, its restoring force is proportional to the stretch. i.e., F = kx where x is the stretch. For a jumper of mass m, we use a conservation principle to determine the length of stretch, d, for a cord given its stiffness, k, and the stiffness, k, of a cord given its stretch, d. The latter expression can be used to determine the number of g's experienced by the jumper.
Part 2 - We make the same assumptions as in Part 1 and introduce a drag component for the jumper. We then model the rise and fall of the jumper over time.
Part 1 - The gravitational potential energy of the jumper isFG = m*g*(L + d)and the elastic potential energy of the cord (spring) isFE = k*d2/2Energy considerations dictate that gravitational potential energy of the jumper in the initial state (top of the jump) is equal to elastic potential energy of the cord in the final state (bottom of the jump); so,m*g*(L + d) = k*d2/2This equation can be solved for d given m, g, k, and L or for k given m, g, d, and L.
As an example, if a given jump height (L + d) is to be matched with a given person (m), then the stiffness of the cord, k, will be determined byk = 2*m*g*(L + d)/d2The maximum force exerted on the jumper is then F = k*d and the ratio of F to the jumper's weight (m*g) produces the acceleration in g's to which the jumper has been subjected. See the Ref., pg. 4, for more details.
Part 2 - In this part, we develop a one-dimensional model governing the time history of the jumper's rise and fall. Our model is discrete, meaning we will monitor the jumper's position and velocity at equally-spaced values in time; let this time spacing be dt. Initially, we assume that the jumper is at x0 = h meters above the ground with initial velocity v0 = 0. Assume that we are at some point in time, told, with position, xold, and velocity, vold, and want to advance to a new point in time, tnew = told + dt. The following equations can then be used to find xnew and vnew:xnew = xold + vold*dtThe coefficient, c, introduces drag into the model and the quantity, b, defined by
vnew = vold + (- g + b/m - c*abs(vold)*vold/m)*dt
tnew = told + dtb = 0 if (h - xold) <= L orinsures that the cord does not affect the jumper's rise or fall until the jumper is at least a distance L from his/her jumping off point.
b = k*(h - L - xold) if (h - xold) > L.
To implement these equations, we setxold = x0, the initial height of the jumperand solve for xnew and vnew at tnew = told + dt. We then set
vold = v0, the initial velocity of the jumper.
told = 0xold = xnew, vold = vnew, and told = tnewand repeat the process until we reach a designated final time.
Part 1 - Given a cord length (L) and a jumper (m), study the relationship between cord stretch, d, and cord stiffness, k, and the resulting g-force exerted on the jumper. This would be useful information in designing a bungee jumping facility. As an example, an interesting jump site might be the Taos Gorge Bridge. It's about 200 m high. If your bungee cord were 50 m long, this would allow about 150 m stretch room. Select 84 kg for your mass and see what you come up with. What is a safe g-level for the jumper?
Part 2 - Solve the equations defining the position-velocity history as the jumper falls and rises under the influence of the cord, gravity, and air resistance. Let dt = 0.01; a reasonable value for c, our drag coefficient, would be c = (m*g)/552 where 55 m/sec is the terminal velocity for a jumper in free fall. Solve from t = 0 to t = 60 sec (1 minute). A "for loop" and an "if-then-else structure" will be useful constructs in your programming logic.
How might you make your simulation more realistic? See the Ref. for some suggestions.