A bungee jumper jumps from a bridge with a 3 m long bungee cord. Once the cord starts to stretch, it acts like an ideal spring with coefficient of elasticity k=100 N/m. What should be the minimum height of the bridge in m so that this jump is safe for the jumper?
Details and assumptions
The jumper has a mass of 80 kg.
The acceleration of gravity is −9.8 m/s^2.
There are at least two ways to solve this problem.
Treat the jumper as a point mass at the end of the cord.
i think we will get from your way damon h be about 15.68 which is wrong..
no no your correct but correct answer will bet putting h-3 in place of h..
The correct answer is 21.276
Consider the law of conservation of energy in solving this problem.
i.e. gravitational potential energy = elastic potential energy.
So, m g (h-3) = 1/2 k x^2.
To solve this problem, we need to consider the forces acting on the bungee jumper.
Let's start by calculating the gravitational force acting on the jumper. The force of gravity can be calculated using the formula:
F_gravity = mass * acceleration due to gravity
Where:
mass = 80 kg (mass of the bungee jumper)
acceleration due to gravity = -9.8 m/s^2 (negative sign indicates downward direction)
Substituting the values, we get:
F_gravity = 80 kg * (-9.8 m/s^2)
Next, we need to calculate the maximum elastic force exerted by the stretched bungee cord. The elastic force can be calculated using Hooke's Law:
F_elastic = k * x
Where:
k = 100 N/m (coefficient of elasticity of the bungee cord)
x = extension or displacement of the spring (in this case, the stretched length of the bungee cord)
We know that the bungee cord stretches 3 m, so x = 3 m.
Substituting the values, we get:
F_elastic = 100 N/m * 3 m
Now, we can determine the minimum height of the bridge needed for a safe jump. To ensure a safe jump, the maximum elastic force exerted by the cord should be greater than or equal to the force of gravity.
So, we need to find the height at which the elastic force equals the gravitational force. Therefore, we set F_elastic equal to F_gravity:
F_elastic = F_gravity
100 N/m * 3 m = 80 kg * (-9.8 m/s^2)
Simplifying, we find:
300 N = -784 N
Since the equation results in a negative value, it means that the height is not high enough to make the jump safe.
To find the minimum height, we can divide the right-hand side of the equation by -9.8 m/s^2 (the acceleration due to gravity):
300 N / (-9.8 m/s^2) = (80 kg * -9.8 m/s^2) / (-9.8 m/s^2)
Simplifying, we get:
30.61 m = 80 kg
Therefore, the minimum height of the bridge should be approximately 30.61 meters to ensure a safe jump for the bungee jumper.
potential energy decrease in fall = potential energy put into spring
m g h = (1/2) k h^2
h = 2 m g /k