Physics of bungee jumping

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Source: http://www.flickr.com/photos/matt-dinnery/3849606743The physics of bungee jumping is an exciting subject of analysis. The standard physics behind this activity is self-apparent. The bungee jumper jumps off a tall framework such as a bridge or crane and also then falls vertically downward till the elastic bungee cord slows his descent to a soptimal, before pulling him earlier up. The jumper then oscillates up and also dvery own till all the energy is dissipated.However, what is specifically amazing in the adhering to analysis of the physics of bungee jumping is that the jumper experiences a downward acceleration that exceeds free-autumn acceleration as a result of gravity. This takes place in the initial component of the autumn while the bungee cord is sabsence (i.e. not stretched). The physics taking place below will be examined next.Free-Fall Acceleration Greater Than gThe following schematic for this evaluation reflects a simplified representation of a bungee jumper and also bungee cord, at the initial position (1), before he jumps. The jumper is stood for as a allude object of mass M. The bungee cord is represented by two lengths of rope, each through length L/2, through a bfinish at the bottom of radius R. The left side of the bungee cord is attached to a resolved support. The mass of the bungee cord is m. The acceleration as a result of gravity is g (equal to 9.8 m/s2 on earth). The authorize convention is "up" as positive and also "down" as negative.Further assumptions in this analysis are:• The jumper M falls vertically downward in the time of the loss, and the falling component of the bungee cord stays straight below him throughout the initial component of the autumn wbelow the bungee cord is sabsence.• Friction and air resistance deserve to be neglected.• The radius R of the bend is little loved one to the size of the directly sections of the bungee cord. Therefore, the length of the bungee cord is roughly L.• During the initial component of the autumn, the added stretching that occurs in the hanging part of the bungee cord as it supports more of its own weight, is negligible. This means that the adjust in elastic potential power of the hanging component of the bungee cord (the left side of the cord in the two figures below) is small enough to be neglected in the conservation of power equation. This is equation (1) in the analysis given listed below.

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The adhering to schematic for this analysis reflects a representation of the bungee jumper and bungee cord after he jumps. This is delisted as position (2). The position of jumper and cord is collection as a role of y which is the position of the jumper M family member to the datum (liked as his original vertical position). The lengths of the straight sections of the cord are provided as a function of y, and also are based on the geomeattempt of the trouble. The vertical velocity of the jumper is provided as v, and the vertical acceleration is provided as a. By geometry, the vertical velocity of the bottom guideline of the bend is v/2, and also the vertical acceleration of the bottom guideline of the bfinish is a/2.
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Because friction and also air resistance are neglected, the physics arising between positions (1) and (2) have the right to be analyzed using conservation of power for the mechanism, which consists of bungee jumper and bungee cord. This will certainly permit us to recognize the velocity v of the jumper as a function of y. Due to the fact that friction and also air resistance are neglected, the just force that does job-related on the mechanism is gravity.The equation for conservation of power is offered as follows:
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Where:T1 is the kinetic energy of the bungee jumper and also bungee cord, at position (1)V1 is the gravitational potential power of the bungee jumper and also bungee cord, at position (1)T2 is the kinetic power of the bungee jumper and bungee cord, at place (2)V2 is the gravitational potential power of the bungee jumper and bungee cord, at position (2)Due to the fact that the mechanism is at rest at position (1) the kinetic energy is
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The gravitational potential power of the mechanism at position (1) is given by the weight of the bungee cord (mg) multiplied by the vertical position of its center of mass, as measured from the datum. (Note that the bungee cord is assumed to have uniform density, so the center of mass lies at its midpoint).The bungee jumper M has actually zero gravitational potential power bereason he is situated at the very same height as the datum.Therefore,
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Keep in mind that this value is negative because the facility of mass of the bungee cord is listed below the datum.For convenience set the thickness of the bungee cord as ρ which is identified as mass per unit length. This is given by
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The kinetic power of the system at place (2) is
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The initially term on the best is the kinetic power of the right area of bungee cord listed below the jumper (M). This section of cord is relocating at velocity v. The second term on the right is the kinetic energy of the jumper M, that is likewise relocating at velocity v.The gravitational potential energy of the mechanism at position (2) is given by
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Once aget, store in mind that we are ignoring the fairly tiny mass of bungee cord at the bottom of the bend. Because of this this mass does not show up in the expressions for kinetic and also potential power given above.Substitute equations (2)-(5) into equation (1) and also settle for v. We get
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This expression conveniently relates the velocity v of the bungee jumper to his vertical position y. Once aget, save in mind that y The next step in this analysis is to use the principle of impulse and momentum to deal with for the acceleration of the bungee jumper (a).To erected this evaluation we shall isolate part of the mechanism making use of a regulate volume type of analysis, which encloses the part of the mechanism (consisting of mass M and bungee cord) that we wish to analyze. This addressed control volume is cleverly chosen in order to more conveniently recognize the acceleration a of the bungee jumper, as he drops.

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In this liked regulate volume the bungee cord deserve to be imagined as "flowing" across the system boundary (stood for by the dashed line) at the lowest part of the bfinish, where the bungee cord stress and anxiety is horizontal. The vertical component of pressure at this area is assumed to be exceptionally tiny, which is an excellent assumption for ropes and rope-favor structures.Due to the fact that only the vertical pressures acting on this system (enclosed by the regulate volume) will certainly impact a, the stress T does not need to be well-known. This saves computation time.The vertical pressure acting on the device enclosed in the manage volume is gravity, which have the right to quickly be accounted for.The next action in this analysis is to use Calculus to set up the governing equations.Consider the figure below. The two stages, (1) and also (2), show the "state" of the device at time t and also time t+dt, where dt is a really little (infinitesimal) time step.Once aget, we are ignoring the mass of the rope segment of radius R at the bottom of the bfinish.
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Between (1) and (2), the adjust in direct momentum in the y-direction of all the pshort articles in the system (identified by the manage volume), is due to the amount of the outside pressures in the y-direction acting on all the particles in the mechanism.We can express this mathematically utilizing Calculus and the principle of impulse and momentum:
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Where:mcv is the mass of the pwrite-ups inside the manage volume, at stage (1)ΣFy is the amount of the external forces in the y-direction acting on all the pposts in the regulate volumedmcv is the change in the mass of the pwrite-ups inside the control volume, between (1) and (2). Note that dmcv dmp is the mass of the pposts that have "flowed" to the exterior of the regulate volumedv is the readjust in vertical velocity of the particles inside the regulate volume, in between (1) and also (2). Keep in mind that (ignoring the bend) all the pshort articles inside the control volume relocate at the exact same vertical velocityExpand the above expression. In the limit as dt→0 we might ignore the 2 "second-order" terms dmcvdv and also dmpdv/2. Divide by dt and also simplify. We get
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Now, the mass of the pshort articles flowing out of the control volume should be a positive amount. Thus,
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This implies that
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Substitute the above equation into equation (7) and we get
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Now,
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Substitute this right into the previous equation and we get
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Now,
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The amount of the outside pressures in the y-direction acting on the pshort articles of the manage volume is the gravitational force. This is offered by
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Substitute equations (9)-(11) right into equation (8) and also we get
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Solve for a. This gives us
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As you can see, the acceleration a is higher than g.Note that this equation is only valid in between y = 0 and also y = -L.To acquire an expression for a in regards to y substitute equation (6) right into equation (13). This offers us
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Once aacquire, y For convenience collection μ = m/M. The over equation then becomes
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The position y = -L is the allude where acceleration a is maximum. For example, if the bungee cord weighs the exact same as the bungee jumper the maximum acceleration is roughly 1.6g.If we substitute a = d2y/dt2 into the over equation we achieve an ordinary differential equation which we can settle numerically to identify y as a duty of time. The solution is subject to the initial problems at time t = 0. These initial problems are: y = 0 and dy/dt = v = 0 (the mechanism begins falling from rest).Even though this evaluation is simplified, the result that a > g renders feeling because the change of velocity (from v to v/2) of the bungee cord as it travels about the bfinish means that tright here have to be tension in the cord that "pulls" upward. This tension deceleprices the cord at the bend place. This pull consequently, reasons the segment of cord above the bend, and the jumper, to accelerate downwards faster than g.In the final evaluation let's look at the maximum distance the bungee jumper drops. Maximum Falling Distance
Once the bungee jumper falls a distance y = -L, the bungee cord loses slack and also tigh10s. This forceful transition is a type of inelastic collision between bungee jumper and cord, and also need to therefore be accounted for in this analysis in order to make specific predictions. To deal with for the velocity of the jumper instantly after the cord pulls tight, one needs to experimentally determine just how much power is shed in the time of the "collision". Once this brand-new velocity is calculated, the conservation of power can as soon as even more be applied in order to recognize the maximum falling distance of the jumper.However before, for illustrative functions this power loss will be ignored, and we shall apply conservation of energy to determine just how much the bungee jumper drops, based upon his initial position prior to jumping.This evaluation is put up making use of the schematic shown below. Once even more we are ignoring friction, air resistance, and the relatively little mass of bungee cord at the bottom of the bend.
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Where:L is the (unstretched) length of the bungee cords is the amount the bungee cord has actually stretchedG is the center of mass of the bungee cordk is the spring continuous of the bungee cord, which is assumed to behave as a linear elastic springThe maximum distance the bungee jumper drops coincides to the lowest point in the autumn, wright here the velocity of the mechanism is zero. This indicates that the kinetic energy of the device at the lowest point is zero.Thus, we deserve to put up the conservation of power equations for the system, comparable to prior to, wbelow place (1) corresponds to the initial (rest) position and position (2) coincides to the lowest point in the loss (wright here kinetic power is zero). The just extra consideration is that we must likewise account for the potential energy of the bungee cord as it stretches.Now,
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Note that the last term in the above equation represents the potential power of the bungee cord, which is assumed to behave actually as a direct elastic spring.Substitute equations (15)-(18) into equation (14). You deserve to then resolve for s numerically or via the quadratic roots formula.For instance, if k = 500 N/m, L = 15 m, g = 9.8 m/s2, and m = M = 60 kg, the amount of stretch of the bungee cord is s = 8.6 m. Now, because some power is actually shed once the bungee cord loses sabsence and pulls tight, this amount of stretch is a bit better than it would be in real life.This concludes the bungee jumping physics evaluation.Rerevolve to The Physics Of Sports
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