Solve Bungee Jump Issue: Derive Expression for x Double Dot, Show mg >= kh/4

[Aihara]

Homework Statement

Typical bungee jumper. Rope has natural length h, spring constant k, man has mass m.

The safety limit for the max acceleration of the man is 3g. Derive an expression for x double dot, and show mg *greater or same than* kh/4

Homework Equations

Well I'm sure everyone knows them
I'm confused about Elastic Energy (kx^2) and Elastic Potential Energy (kx^2/2l), which do I use?

The Attempt at a Solution

I'm got four pages of working but not get where I want.
The acceleration (a) isn't constant, so are we trying to get an expression for a in terms of time or string extension...

At a random point after the rope has started extending mg - T = ma. so mg - kx/h = ma
so rearrange for acceleration... but then the x won't cancel and I can't show what I was asked too...

Thanks

 

[tiny-tim]

Welcome to PF!

At a random point after the rope has started extending mg - T = ma. so mg - kx/h = ma
so rearrange for acceleration... but then the x won't cancel and I can't show what I was asked too...

Thanks

Hi Aihara! Welcome to PF!

(You won't need energy, only force)

why should the x cancel?

a = x'', so you should get an equation relating x'' and x …

then solve it.

 

[thomate1]

for providing the necessary information and trying to solve the problem. I would approach this problem by first understanding the physical principles involved. In this case, we are dealing with the motion of a bungee jumper, which can be described using Newton's laws of motion and the concept of elastic potential energy.

To derive an expression for x double dot (the second derivative of the displacement x), we can use the equation for elastic potential energy: PE = 1/2 kx^2, where k is the spring constant and x is the displacement of the bungee cord from its natural length.

We can set up the equation of motion for the bungee jumper as follows:

m*x double dot = mg - kx

Where m is the mass of the bungee jumper, g is the acceleration due to gravity, and kx is the force exerted by the bungee cord on the jumper.

To solve for x double dot, we can rearrange the equation as follows:

x double dot = (mg - kx)/m

Now, to ensure the safety of the bungee jumper, we need to make sure that the acceleration (x double dot) does not exceed 3g. This means that:

x double dot <= 3g

Substituting the expression for x double dot derived above, we get:

(mg - kx)/m <= 3g

Simplifying this inequality, we get:

mg <= 3mg - 3kx

Dividing both sides by 3 and rearranging, we get:

mg/3 <= kx

Since we know that the natural length of the bungee cord is h, we can substitute h for x and rearrange to get:

mg/3 <= kh/3

Finally, dividing both sides by 4, we get:

mg/4 <= kh/4

This shows that the mass of the bungee jumper multiplied by the acceleration due to gravity (mg) must be greater than or equal to the product of the spring constant (k) and the natural length of the bungee cord (h) divided by 4. This ensures that the acceleration of the bungee jumper does not exceed 3g, as required for safety.

In conclusion, we can use the equation for elastic potential energy to derive an expression for x double dot and then use this expression to show that mg >= kh/4 for the safety