## A Bungee Jumper with Mass 75 kg has a Bungee Cord
A bungee jumper with mass 75 kg has a bungee cord with a spring constant of 100 N/m.
### The following are the steps on how to calculate the maximum height of the jump:
1. Calculate the potential energy stored in the bungee cord at its maximum stretch. This is equal to 1/2 * spring constant * (change in length)^2.
2. The bungee jumper’s potential energy at its starting height is equal to its mass * acceleration due to gravity * initial height.
3. Equate the potential energy stored in the bungee cord to the potential energy of the bungee jumper at its starting height.
4. Solve the resulting equation for the initial height.
### Calculations:
1. Potential energy stored in the bungee cord:
“`
PE = 1/2 * k * 螖x^2
“`
where:
– k = spring constant = 100 N/m
– 螖x = change in length of the bungee cord
Assuming the bungee cord was stretched to twice its original length:
– 螖x = 2 * (original length) = 2L
Therefore:
“`
PE = 1/2 * 100 N/m * (2L)^2 = 200 L^2 J
“`
2. Potential energy of the bungee jumper at its starting height:
“`
PE = mgh
“`
where:
– m = mass of the bungee jumper = 75 kg
– g = acceleration due to gravity = 9.81 m/s^2
– h = initial height
Therefore:
“`
PE = 75 kg * 9.81 m/s^2 * h = 735.75 h J
“`
3. Equating potential energies:
“`
200 L^2 J = 735.75 h J
“`
4. Solving for h:
“`
h = (200 L^2 J / 735.75 J) m
“`
Assuming the original length of the bungee cord was 10 m:
“`
h = (200 * 10^2 m^2 / 735.75 m^2) m
“`
“`
h 鈮
“`
Therefore, the maximum height of the bungee jump is approximately 27.20 meters.
### Factors Affecting Maximum Height:
– Mass of the jumper: A heavier jumper will stretch the bungee cord less, resulting in a lower maximum height.
– Spring constant of the bungee cord: A bungee cord with a higher spring constant will stretch less for the same force, resulting in a lower maximum height.
– Initial length of the bungee cord: A longer bungee cord will store more potential energy at its maximum stretch, resulting in a higher maximum height.
– Air resistance: Air resistance will act as a force opposing the upward motion of the jumper, reducing the maximum height.