/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 2 The water passing over Victoria ... [FREE SOLUTION] | 91Ó°ÊÓ

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The water passing over Victoria Falls, located along the Zambezi River on the border of Zimbabwe and Zambia, drops about \(105 \mathrm{m} .\) How much internal energy is produced per kilogram as a result of the fall?

Short Answer

Expert verified
The internal energy produced per kilogram of water is approximately 1030.05 J/kg.

Step by step solution

01

Understand the Problem

The water falling from a height will convert its potential energy into internal energy due to gravity. We need to calculate the internal energy produced per kilogram of water as it falls 105 meters.
02

Know the Formula for Potential Energy

Potential energy (PE) is given by the formula \(PE = mgh\), where \(m\) is mass, \(g\) is the acceleration due to gravity, and \(h\) is the height.
03

Simplify for Energy per Kilogram

Since we need the energy per kilogram, we can simplify the formula to \(PE = gh\), assuming \(m = 1 \text{ kg}\). This is because we will calculate the energy produced for 1 kilogram of water.
04

Use the Gravitational Constant

In most physics problems, the gravitational constant \(g\) is approximately \(9.81 \text{ m/s}^2\). Use this value for calculations.
05

Calculate Internal Energy Per Kilogram

Substitute the values into the formula: \[ PE = gh = 9.81 \, \text{m/s}^2 \times 105 \, \text{m} \]This yields: \[ PE = 1030.05 \, \text{J} \]
06

Conclude the Calculation

The potential energy, which converts to internal energy per kilogram of water, is approximately \(1030.05 \text{ J/kg}\). Therefore, this is the amount of internal energy produced per kilogram as the water falls.

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Key Concepts

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

Gravitational Potential Energy
Gravitational potential energy is an essential concept in physics that describes the energy an object has due to its position in a gravitational field. Imagine holding an apple above the ground; it has potential energy because of its elevated position. This energy can be described using the formula:
  • \( PE = mgh \)
where:
  • \(m\) represents the mass of the object
  • \(g\) is the acceleration due to gravity
  • \(h\) refers to the height above a reference point
When the apple falls, this potential energy gets converted into kinetic energy. Similarly, when water cascades over a waterfall like Victoria Falls, the energy transformations follow the same principles.
Internal Energy Conversion
As the water plummets over the majestic Victoria Falls, a fascinating energy transformation occurs. The potential energy at the top of the fall is not lost but rather changes form. As the water descends, its potential energy transforms into internal energy. This could include thermal energy, resulting from the friction and turbulence as the water hits the rocks and water below, and kinetic energy is briefly increased before being dissipated. This illustrates the law of conservation of energy, which states that energy cannot be created or destroyed but only changed from one form to another. In real-world scenarios like this, the energy transformations are complex but can be predicted by understanding fundamental physics concepts.
Energy per Kilogram Calculations
When we discuss energy in terms of per kilogram, it becomes easier to standardize calculations, especially when dealing with fluids like water. By calculating energy per kilogram, we can focus on the work done on a single unit mass, simplifying our measurements.For the water at Victoria Falls, simplifying calculations involves using a mass of 1 kilogram while considering the height of the fall. This leads to using a modified version of the potential energy formula where:
  • \( PE = gh \)
This simplification makes it easier to understand the energy conversion per kilogram of water, yielding clear, interpretable results.
Victoria Falls Physics
Victoria Falls is not just a breathtaking natural wonder; it's also a perfect example to apply physics concepts. Located on the border between Zimbabwe and Zambia, the falls present an extraordinary setting for studying gravitational effects and energy transformations. When water falls from the height of approximately 105 meters, it undergoes significant energy changes. The potential energy at the top is maximum, converting to internal energy at the bottom. The scale and power of Victoria Falls provide a dynamic illustration of how physics operates in nature on a grand scale. Such examples help in visualizing basic physics concepts and their real-world applications.
Gravitational Constant
The gravitational constant \(g\) is a fundamental value in physics, representing the acceleration due to gravity at Earth's surface. This value is approximately \(9.81 \ ext{m/s}^2\), although it can slightly vary based on location.In physics problems, this constant helps calculate forces and energies involving gravity. It's crucial when determining potential energy, as we rely on this value to understand how gravity influences objects. Without a consistent gravitational constant, calculations like those for Victoria Falls would become inaccurate.By using \(9.81 \ ext{m/s}^2\) in our calculations, we can reliably predict how gravitational forces will affect falling water, illustrating the consistent power of gravity on Earth.

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Most popular questions from this chapter

For a cheetah, \(70.0 \%\) of the energy expended during exertion is internal work done on the cheetah's system and is dissipated within his body; for a dog only \(5.00 \%\) of the energy expended is dissipated within the dog's body. Assume that both animals expend the same total amount of energy during exertion, both have the same heat capacity, and the cheetah is 2.00 times as heavy as the dog. (a) How much higher is the temperature change of the cheetah compared to the temperature change of the dog? (b) If they both start out at an initial temperature of \(35.0^{\circ} \mathrm{C},\) and the cheetah has a temperature of \(40.0^{\circ} \mathrm{C}\) after the exertion, what is the final temperature of the dog? Which animal probably has more endurance? Explain.
The amount of heat generated during the contraction of muscle in an amphibian's leg is given by $$Q=0.544 \mathrm{mJ}+(1.46 \mathrm{mJ} / \mathrm{cm}) \Delta x$$ where \(\Delta x\) is the length shortened. If a muscle of length \(3.0 \mathrm{cm}\) and mass \(0.10 \mathrm{g}\) is shortened by $1.5 \mathrm{cm}$ during a contraction, what is the temperature rise? Assume that the specific heat of muscle is $4.186 \mathrm{J} /\left(\mathrm{g} \cdot^{\circ} \mathrm{C}\right)$.
It takes \(880 \mathrm{J}\) to raise the temperature of \(350 \mathrm{g}\) of lead from 0 to \(20.0^{\circ} \mathrm{C} .\) What is the specific heat of lead?
Small animals eat much more food per kg of body mass than do larger animals. The basal metabolic rate (BMR) is the minimal energy intake necessary to sustain life in a state of complete inactivity. The table lists the BMR, mass, and surface area for five animals. (a) Calculate the BMR/kg of body mass for each animal. Is it true that smaller animals must consume much more food per kg of body mass? (b) Calculate the BMR/m" of surface area. (c) Can you explain why the BMR/m \(^{2}\) is approximately the same for animals of different sizes? Consider what happens to the food energy metabolized by an animal in a resting state.

How much internal energy is generated when a \(20.0-\mathrm{g}\) lead bullet, traveling at \(7.00 \times 10^{2} \mathrm{m} / \mathrm{s},\) comes to a stop as it strikes a metal plate?

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