Question

A bungee jumper (mass 80 kg) is attached to a 10 m bungee cord attached to the top of a crane. For the first 5 meters of extension, the force exerted by the bungee cord increases by 2 N for every 1 cm. For any further extension, the force increases by only 1.2 N for every 1 cm..

a) Sketch the elastic force vs the length of the bungee cord.

b) What is the maximum extension of the bungee cord in this situation?

Answer 1

a)

b) The jumper will be falling downward with an acceleration of g until the bungee chord doesn't reaches its natural length.

Thus when the chord reaches its natural length the velocity of the jumper will be

v = (2gh)1/2

here h = 10m as the chord has to attain its natural length

v = (2 x 9.8 x 10)1/2

=> v =14 m/s

Now the jumper will start experiencing the force due to elastic extension, thus it will try to reduce the acceleration due to gravity.

Since for the first 5m of extension we have k =2N/cm = 200N/m

We use the work-energy theorem to find the velocity for extension till 5m

1/2mvf2 - 1/2mvi2 = mgh - 1/2kx2

=> vf = (vi2 + 2gh -kx2/m)1/2

Now since h = x = 5m,

vf = 15.215 m/s

after this 5m extension the chord will extend with a k of 1.2N/cm = 120N/m

Now again using the work-energy theorem to find the extension till which the final velocity becomes zero we get,

=> 1/2mvf2 - 1/2mvi2 = mgh - 1/2kx2

=>  - 1/2mvi2 = mgh - 1/2kx2

Now the x here will turn to x = 5+h as the chord has already been extended for 5m

=> -vi2 = 2gh - k(5+h)2/m

=> 1.5(5+h)2 - 19.6h - 231.5 = 0

Now solving for h we get,

h = 13 and h = -9.942 ,

So h = 13m

So we have net maximum extension in the string as x = 5m + 13m = 18m