A person is planning a bungee jump from a 40 meter high bridge. Under the bridge is a river with crocodiles, so the person does not want to be submerged into the water. The rubber rope fastened to the ankles has a spring constant of 3,600 N/m per meter length. Thus one meter length of rope will have a spring constant of 3,600 N/m, two meters of rope will have a spring constant of 1,800 N/m, three meters of rope will have a spring constant of 1,200 N/m and so forth. The distance from the ankles to the top of the head is 175 cm, and the person has a mass of 80 kg. Calculate: a. b. c. how long the rope should be [m] how much deflection the rope experiences [m] how much impact force the bungee jumper experiences [N]

a. l = 19.7m, b. 18.55m, c. Impact Force = 3889.84 N

Explanation:

The total energy of the system when the person is at top of the bridge is

Potential energy = mgh, Kinetic energy = 0

The total energy of the system when the person reaches just above the surface

Potential energy = 0, Kinetic energy = 0, Spring energy = ½ K X2, where k is the spring constant and X is the deflection

Applying conservation of energy

mgh = 0 + 0 + ½ K X²

80 x 9.81 x 40 = ½ (3600/l) X²

31392 = ½ (3600/l) X²

We can also conclude that

l + X + 1.75 = 40

l + X = 38.25

a. Substitute the value of x from above into the energy conversion expression

31392 = ½ (3600/l) (38.25 - l) ²

31392 x 2/3600 = (38.25 + l² - 2l (38.25)) / l

17.44l = l2 - 76.5l + 38.25²

l² - 76.5l - 17.44l + 1463.0625 = 0

Solving for l we get

L = 19.7

Hence, length of the rope is 19.7m

b. The deflection is calculated by using the relation between l and X

L + X = 38.25

X = 38.25 - 19.7 = 18.55m

c. The impact force is calculated using the impact force formula which relates the impact force with the deflection

F = KX

F = (3600/l). X

F = (3600/19.7). (18.55) = 3889.84 N

Thus, the impact force is 3889.84 N