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Chapter 7: Problem 15

A \(60-\mathrm{kg}\) hiker ascending \(1250-\mathrm{m}\) -high Camel's Hump mountain in Vermont has potential energy \(-240 \mathrm{kJ} ;\) the zero of potential energy is taken at the mountaintop. What's her altitude?

Short Answer

Expert verified
The altitude of the hiker is about 4081.633 meters below the mountaintop.

Step by step solution

01

Write down the potential energy equation

The potential energy (\(PE\)) of an object of mass \(m\) at a height \(h\) can be given by: \[PE = m \cdot g \cdot h\] where \(g = 9.8 \, m/s^2\) is the acceleration due to gravity.
02

Calculate the height using the formula

Substituting the given values for \(PE\) and \(m\) into the formula and solving for \(h:\) \[h = \frac{PE}{m \cdot g} \] Therefore, inserting the given values, we get: \[h = \frac{-240 \times 10^{3}}{60 \times 9.8}\]
03

Result

Upon carrying the operation above, we obtain: \[h = -4081.633\,m\] The negative sign indicates that the hiker is below the mountaintop, and thus, 4081.633 meters below the mountaintop.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Potential Energy
Potential energy is the stored energy an object possesses due to its position or state. Imagine stretching a rubber band; the more you stretch, the more potential energy you store in it. In physics, this concept explores how objects have the capacity to do work as a result of where they are or how they're arranged.

In the context of our problem, the potential energy is associated with the hiker's position relative to the mountain. Specifically, we're dealing with gravitational potential energy which is the energy an object has due to its height above a reference point, which, in this case, is the mountaintop.
Gravitational Potential Energy Explored
Gravitational potential energy (GPE) is a form of potential energy that depends on an object's vertical position and mass within a gravitational field, such as that of Earth. The equation \ \( GPE = m \cdot g \cdot h \) \ encapsulates this relationship, where \(m\) is mass, \(g\) is the gravitational field strength (on Earth typically \(9.8 \, m/s^2\)) and \(h\) is the height above a zero-potential reference point.

In the exercise, the hiker's GPE is negative because the reference point, the top of Camel's Hump mountain, is higher than where the hiker stands. The negative sign merely indicates direction—downwards from the reference level.
The Principle of Energy Conservation
Energy conservation is a fundamental concept in physics, stating that within a closed system, energy cannot be created or destroyed, only transformed from one form to another. This law reminds us that the total energy before an event must be equal to the total energy after—although the form of that energy might change, as when potential energy becomes kinetic energy during a fall.

The hiker's GPE will convert to other energy forms (like kinetic energy, if she descends) without any loss of the total energy of the system as she moves relative to the mountain. Her potential energy value at any given altitude is one snapshot of the energy's form in the conservation dance.
Solving Physics Problems Effectively
Physics problem-solving requires a systematic approach: understanding the problem and the principles involved, translating these into mathematics, and performing logical computations. In our scenario, we first grasped what potential energy is (step 1), used the GPE equation to set up our mathematical model (step 2), and then computed the answer with careful attention to the negative sign (step 3), reflecting our reference point choice.

For clear problem-solving, always define your variables, reference points, and signs (-/+). This clarity is essential in interpreting results correctly, as it was in understanding the hiker's position relative to the mountain top in our exercise.

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Most popular questions from this chapter

You have a summer job at your university's zoology department, where you'll be working with an animal behavior expert. She's assigned you to study videos of different animals leaping into the air. Your task is to compare their power outputs as they jump. You'll have the mass \(m\) of each animal from data collected in the field. From the videos, you'll be able to measure both the vertical distance \(d\) over which the animal accelerates when it pushes off the ground and the maximum height \(h\) it reaches. Your task is to find an algebraic expression for power in terms of these parameters.

A tightrope walker follows an essentially horizontal rope between two mountain peaks of equal altitude. A climber descends from one peak and climbs the other. Compare the work done by the gravitational force on the tightrope walker and the climber.

The force exerted by an unusual spring when it's compressed a distance \(x\) from equilibrium is \(F=-k x-c x^{3},\) where \(k=220 \mathrm{N} / \mathrm{m}\) and \(c=3.1 \mathrm{N} / \mathrm{m}^{3} .\) Find the stored energy when it's been compressed \(15 \mathrm{cm}\)

A block of weight \(4.5 \mathrm{N}\) is launched up a \(30^{\circ}\) inclined plane 2.0 \(\mathrm{m}\) long by a spring with \(k=2.0 \mathrm{kN} / \mathrm{m}\) and maximum compression \(10 \mathrm{cm} .\) The coefficient of kinetic friction is \(0.50 .\) Does the block reach the top of the incline? If so, how much kinetic energy does it have there? If not, how close to the top, along the incline, does it get?

A particle with total energy \(3.5 \mathrm{J}\) is trapped in a potential well described by \(U=7.0-8.0 x+1.7 x^{2},\) where \(U\) is in joules and \(x\) in meters. Find its turning points.
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