# Challenging physics question, hooke's law and conservation of energy

• itunescape
In summary, the question involves a bungee jumper wanting to jump a certain distance with a uniform elastic cord. The length of the cord must be less than the total distance the man will jump, and the kinetic energy at the initial point must be zero. The potential energy at the final point must also be zero because the cord can only stretch a certain distance. Using the conservation of energy equation, it can be determined that the effective spring constant of multiple springs or strings connected in series is given by 1/k_eff = ∑(1/ki). This can be used to find the spring constant of the longer cord needed for the jump.
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## Homework Statement

Obeying hooke's law. A hot air balloon is 65.0m from the ground. The bungee jumper wants to jump with a uniform elastic cord up to 10m above the ground. During a preliminary test, the cord at rest was 5.0m and when he got on it stretched 1.50m. a) What length of cord should he use? b)What maximum acceleration will he experience?

(The problem was much longer than this i swear, but above is all that is given).

## Homework Equations

ke(initial) + Pegrav(initial)+Pespring(initial)=Ke(final)+Pegrav(final)+Pespring(final )

1/2mv^2+mgh+1/2kx^2=1/2mv^2+mgh+1/2kx^2
Fs= -kx
F=ma
F=mg

## The Attempt at a Solution

The question is long but only distances are given... I made an attempt to solve for k and mass but it was futile. There are too many variable to solve for, unless by some miracle, the unknowns cancel out.
total D= 65.0m
final D= 10.0m
total D-final D= 55.0m total distance the man will bungee jump.
Preliminary test data:
at rest cord = 5.0m
mass stretched the cord 1.50m
new cord length= 7.50m

-The length of the cord must be less than 55 but needs to stretch to 55m. That I can safely assume.
-The Kinetic energy at the initial point must zero
-The potential energy at the final point must be zero bc the cord can only stretch that far (55 m down).

conservation of energy:initial = final

1/2mv^2+mgh+1/2kx^2=1/2mv^2+mgh+1/2kx^2
0 + m*g*h + 1/2k*x^2= 1/2m*v^2 + 0 + 1/2k*x^2
m*g*h+ 1/2k*x^2=1/2m*v^2 + 1/2k*x^2
plug in what i can assume to be the right #s:

m*9.8*65.0 + 1/2 k*x^2= 1/2m*v^2 + 1/2k*10^2

from here I'm stuck because there is m, v, k to solve for along with x.
So i took a desperate approach >>>
from the data given on the preliminary test within the problem, i took a porportional approach.

5.00m X
------ = -------
7.50m 55.0m

X= 36.7 m (This is not the correct answer, but i think it was a nice try lol)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
My peer's approach:
1/2mv^2+mgh+1/2kx^2=1/2mv^2+mgh+1/2kx^2
k=mg/x
v= sqrt(2*g*h)
in conservation of energy equation factor out the mass.
1/2m*(sqrt2*g*h)^2+mgh+1/2(mg/x)*x^2=1/2m*(sqrt2*g*h)^2+mgh+1/2(mg/x)*x^2
1/2m*(sqrt2*9.8*55)^2 +m*9.8*55+1/2(m9.8/1.5)*1.5^2=1/2m*(sqrt2*9.8*10)^2+m*9.8*10+1/2(m*9.8/x)*x^2

he stopped here because the equation got too messy and made no sense.
I am not asking for an answer, just the logic to solving such a complicated problem. What is the first step suppose to be? solve for mass? does that get canceled out in the end?
Thank you for your time for attempting such a lengthy problem.

itunescape said:
Thank you for your time for attempting such a lengthy problem.

This short problem has been stretched more than the bungee cord...

The salient point to recognize here is that the effective spring constant of multiple springs or strings connected in series is given by 1/k_eff = ∑(1/ki). For example, if two identical cords are joined in series to make a new one double the length, then the new k' is given by:

1/k' = 1/k + 1/k => k' = k/2. This follows directly from Hooke's law.

Using this, the spring constant k2 of the longer cord should be k2 = (L1/L2)K, where L1 and L2 are lengths of the shorter and longer cords respectively. L2 is the unknown to be found, and k1 of the shorter test cord of length L1 is given.

Now equate the initial PE_grav to the final PE_spring.

I hope that is enough for you proceed further.

I understand the frustration of attempting a complex problem with limited information. However, it is important to approach the problem systematically and use the given information to guide your solution.

The first step in solving this problem would be to analyze the forces acting on the bungee jumper at different points in the jump. At the initial point, the only force acting on the jumper is gravity, which is balanced by the tension in the cord. At the final point, the jumper experiences a maximum acceleration due to the tension in the cord and gravity pulling him downwards.

Using Hooke's Law, we can write an equation for the tension in the cord at any point during the jump. This will involve the spring constant, k, and the length of the cord, which can be expressed as a function of the jumper's position. This will give you an equation for the acceleration experienced by the jumper at any point during the jump.

Next, we can use the conservation of energy equation to relate the potential energy and kinetic energy at the initial and final points of the jump. This will involve the mass of the jumper, which can be solved for using the given information about the cord's length and stretch.

Once you have equations for the tension and acceleration, you can use the fact that the jumper wants to reach a maximum height of 10m to solve for the length of the cord that will provide the desired jump.

I would recommend breaking the problem down into smaller steps and solving for one variable at a time. This will help you keep track of the information you have and guide your solution. Also, it may be helpful to draw a diagram of the forces acting on the jumper at different points in the jump to visualize the problem.

In short, the key to solving this problem is to use the given information and equations to systematically solve for the unknown variables. It may seem overwhelming at first, but with careful analysis and organization, you can find a solution.

## 1. What is Hooke's Law?

Hooke's Law is a fundamental principle in physics that describes the relationship between the force applied to an elastic material and the resulting deformation. It states that the force applied is directly proportional to the amount of deformation produced, as long as the material remains within its elastic limit.

## 2. How is Hooke's Law related to conservation of energy?

Hooke's Law is closely related to the principle of conservation of energy. When an elastic material is stretched or compressed, potential energy is stored in the material due to the change in its shape. This potential energy is converted back into kinetic energy when the material returns to its original shape, demonstrating the conservation of energy.

## 3. Can Hooke's Law be applied to all materials?

No, Hooke's Law only applies to materials that exhibit elastic behavior, meaning they can return to their original shape after being deformed. Materials that do not behave elastically, such as plastic or rubber, do not follow Hooke's Law.

## 4. What are some real-world applications of Hooke's Law?

Hooke's Law has many practical applications, such as in the design of springs, shock absorbers, and other mechanical devices. It is also used in the field of materials testing to determine the strength and elasticity of different materials.

## 5. What are some common misconceptions about Hooke's Law?

One common misconception about Hooke's Law is that it only applies to linear systems, meaning the force and deformation are directly proportional. However, Hooke's Law can also be applied to non-linear systems, where the relationship between force and deformation is not linear but can still be described by a mathematical equation. Another misconception is that Hooke's Law is only relevant for small deformations, but it can also be applied to larger deformations as long as the elastic limit of the material is not exceeded.

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