What is the maximum length a bungee rope with a spring constant of 100 Nm−1 can be for an 80kg man to be able to jump from 100m above a river without touching the water?

The key to solving this problem is realising that gravitational potential energy and elastic potential energy need to be equated. The gravitational potential energy of an object of mass m at height h above the Earth is equal to mgh. The elastic potential energy of a spring with spring constant k and extension x is 0.5kx^2. Equating these expressions gives mgh = 0.5kx^2. This assumes that all of the gravitational potential energy of the bungee jumper will be converted to elastic potential energy. To find the maximum extension of the rope, rearrange this expression to find x, giving x = (2mgh/k)^0.5. Substituting in the values given in the question gives the extension of the rope to be x = (2 x 80 x 9.81 x 100/100)^0.5 = 39.62m. To find the maximum original length the rope can be without the jumper touching the water, just subtract this extension from the height at which the jumper is jumping from, giving 100 - 39.62 = 60.38m. Therefore, the rope can be a maximum of 60.38m long for the jumper to be able to jump from 100m above the river and not touch the water.