A bungee cord is 30.0 m long and, when stretched a distance x, it exerts a restoring force of magnitude kx. Your father-in-law (mass 97.0kg ) stands on a platform 45.0 m above the ground,...

A bungee cord is 30.0 m long and, when stretched a distance x, it exerts a restoring force of magnitude kx. Your father-in-law (mass 97.0kg ) stands on a platform 45.0 m above the ground, and one end of the cord is tied securely to his ankle and the other end to the platform. You have promised him that when he steps off the platform he will fall a maximum distance of only 41.0 m before the cord stops him. You had several bungee cords to select from, and you tested them by stretching them out, tying one end to a tree, and pulling on the other end with a force of 390N .

When you do this, what distance will the bungee cord that you should select have stretched?

Asked on by Wiggin42

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mr-mayonnaise's profile pic

mr-mayonnaise | Middle School Teacher | (Level 1) Assistant Educator

Posted on

The best equation to use for solving this problem is Hooke's Law. Hooke's law is based around elasticity of springs and other "elastic" materials such as a bungee cord. The equation uses the force applied to the spring equaling the distance the spring is stretched times the spring constant (or k). Knowing how the son-in-law stretched it before using the tree, we can use his values to find our constant which was not previously mentioned. 

The question mentions the son using 390 Newtons of force to pull the cord. It also mentions the son believing the cord will only pull to 41 m. I am assuming that is the max distance he pulled the string using the force mentioned and he has severely underestimated the weight of his father in law. Using these values gives us the following numbers:

`F = kx`

`390 N= k(41m-30m)`

`(390N)/(11m)=k`

`=k`

``Knowing our spring constant k, we can find the new stretch distance using the weight of the father-in-law for the force. The force can be found by multiplying his weight (97 kg) by the gravitational constant (9.8 m/s*s).

`F=ma`

`F=(97kg)(9.8m/s^2)`

`F = 950.6 N`

So the father in law is being pulled downward by gravity by 950.6 Newtons of force. If we use this equation and our spring constant in Hooke's law once more we find the new stretch distance to be (x):

`F=kx`

`(950.6N)=(35.45 N/m)x`

`x=26.81m`

This is the total length the bungee cord is meant to stretch out added onto the original length of 30m, making the bungee cord stretch to a total 56.81m. This bungee is cord is no good as the father-in-law is only 45 meters off the ground, meaning he will have no other option than to hit the ground. If we want to find a cord that can stop him in so short a time, we need to find a new k value and a cord attached to it. 

Let's assume the cord is still 30m and we want it to stop after stretching 15 meters (or less). Using the father's weight we can use Hooke's law to find the k value we need: 

`F=kx`

`950.6N=k(15m)`

`(950.6N)/(15m)=k`

`63.37 N/m = k`


The bungee cord that SHOULD have been picked would have a k value greater than 63.37 Newtons per meter. 

This question was very vague and I'm still not sure quite what the answer was they were asking for, but I hope this all made sense and can lead you toward a correct result.

Sources:
Wiggin42's profile pic

Wiggin42 | Student, Undergraduate | (Level 2) Valedictorian

Posted on

The distance the cord stretches is 41 - 30 = 11m. 

The potential energy of the cord equals the potential energy of the father-in-law. 

PE cord is like a spring and equals .5kx^2, where x is 11. 

PE father-in-law is due to gravity and equals mgh, where h is 41. 

Set the two equal to each other and you can solve for the spring constant k. 

The restoring force is modeled by F = kx where F is force, k is the spring constant just found, and x is the distance the bungee cord should stretch. You calculated k and F is given so now you can solve for x. 

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