Harness Bungee
Posted on Monday, August 23rd, 2010 at 2:23 amHarness Bungee
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Really Need Help With This Problem?
It's your birthday, and to celebrate you're going to make your first bungee jump. You stand on a bridge 120 m above a raging river and attach a 35-m-long bungee cord to your harness. A bungee cord, for practical purposes, is just a long spring, and this cord has a spring constant of 40 N/m. Assume that your mass is 70 kg. After a long hesitation, you dive off the bridge.
1) How far are you above the water when the cord reaches its maximum elongation?
*NOTE: The answer is NOT 67.85m or 68 m rounded for some reason. I thought it was but it's not the right answer.
x = mg/k = (70)(9.8)/40
x = 17.15
d = 35 + 17.15 = 52.15
d = 120 - 52.15 = 67.85
This is the distance above the water when you come to a complete stop. However this is not the maximum elongation of the cord.
Think about energy conservation. In this case the change in the gravitational potential energy must equal the elastic energy stored in the cord when the jumper reaches the lowest point.
Change in gravitational potential energy = (change in height)x(weight of jumper)
GPE = H*(70*g) = H*70*9.8 = H686
Elastic energy in cord = (1/2)kX^2
Where X is the change in length of the cord at it's maximum extension.
H686 = (1/2)kX^2 = (1/2)40X^2 = 20X^2
But H is equal to the length of the cord plus the change in length X. So: H = 35 + X
686(35 + X) = 20X^2
20X^2 - 686X - 24010 = 0
X = (686 +/- SQRT((686)^2 + 4(20)(24010))) / (2*20)
X = (686 + 1546)/(40)
X = 55.81
The cord will be extended by 55.81 meters when the jumper is at maximum elongation of the cord. He then oscillates back up and down until eventually reaching equilibrium at 67.85 meters above the water.
Distance above water = 120 - (35 + 55.81)
Distance above water = 29.19 meters
