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The equation that describes the kinetic energy of a moving particle is \(K=\frac{1}{2} m v^{2}\), where \(m\) is the mass of the particle and \(v\) is its speed. Show that 1 joule \((J)\), the SI unit of energy, is equivalent to \(1 \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}^{2}\).

Short Answer

Expert verified
1 Joule (J), the SI unit of energy, is indeed equivalent to \(1 \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}^{2}\).

Step by step solution

01

Understand the Kinetic Energy Equation

The given kinetic energy equation is \(K=\frac{1}{2} m v^{2}\), where K is kinetic energy, m is mass, and v is speed.
02

Express Kinetic Energy in SI Units

The SI unit of kinetic energy is Joules (J), for mass is kilograms (kg), and for speed is meters per second (m/s). Substitute these units into the kinetic energy equation, thus getting \(K= J = \frac{1}{2} (kg) \cdot (m/s)^{2}.\)
03

Simplify the Equation

If you simplify the equation, you will get \( J = kg \cdot m^{2} / s^{2}\) . This shows that 1 Joule is equivalent to \( kg \cdot m^{2} / s^{2}\).

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Key Concepts

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

SI Units
SI Units, or the International System of Units, form the foundation for scientific measurements worldwide. These units ensure consistency and precision in various fields such as science, engineering, and technology.
The base unit for mass in the SI system is the kilogram (kg), for distance or length is the meter (m), and for time is the second (s). These three units are crucial in defining other derived units, such as the Joule for energy, which we'll explore in more detail.
Using the SI system helps scientists communicate their findings without misunderstandings, as these units are universally accepted. It also simplifies conversions within different measurement systems, allowing for more accurate experiments and calculations.
Joule
The Joule is the SI unit of energy, named after the physicist James Prescott Joule, who made significant contributions to the study of energy. Energy, in a broad sense, is the ability to perform work or cause change. In physics, this is quantified in Joules when measuring energy.
One Joule is the energy transferred when one kilogram mass is moved by a force of one newton over a distance of one meter. Mathematically, this can be expressed and justified through the equation for kinetic energy: \[K = \frac{1}{2} m v^2\]Using this equation and substituting the respective SI units for mass (kg) and speed (m/s), we can simplify to show that 1 Joule equals \( kg \cdot m^2 / s^2 \). This demonstrates the relationship between these basic physical quantities when calculating energy.
Energy Conversion
Energy conversion is a common concept that involves changing energy from one form to another. Understanding how energy converts and conserves in different processes is key to various applications, from powering mechanical devices to processing electronic data.
For example, kinetic energy is the energy of movement. It can be converted into potential energy, thermal energy, or other forms during a process. The SI unit for all types of energy remains the Joule, which helps maintain a standard when calculating conversions.
In many scenarios, especially when working with energy transformations in physics, you use the fundamental \[K = \frac{1}{2} m v^2\]formula to compute energy in its kinetic form. Understanding these principles aids in solving real-world problems, such as how much fuel is needed for a car to reach a specific speed.
Mass and Speed
Mass and speed are two fundamental components of kinetic energy. Understanding their roles in the kinetic energy equation helps in comprehending how motion affects energy. In the kinetic energy formula, \( K = \frac{1}{2} m v^2 \), mass \((m)\) is the quantity of matter in an object, measured in kilograms (kg). It signifies how much inertia an object possesses, impacting how it moves and reacts to forces.
Speed \((v)\), measured in meters per second (m/s), denotes how fast an object is moving. Squaring speed in the equation highlights its exponential impact on energy. In simple terms, doubling the speed of an object results in four times the kinetic energy.
Therefore, both mass and speed directly influence the kinetic energy of a moving object, making them vital when calculating and predicting energy changes in various systems.

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Most popular questions from this chapter

Write the following numbers using decimals: A. \(4.42 \times 10^{-3}\) B. \(7.09 \times 10^{-6}\) C. \(8.28 \times 10^{2}\) D. \(6.02 \times 10^{6}\) E. \(456 \times 10^{3}\) F. \(22.4 \times 10^{-3}\) G. \(0.375 \times 10^{-4}\) H. \(138 \times 10^{-6}\)

Medical The concentration of PSA (prostate-specific antigen) in the blood is sometimes used as a screening test for possible prostate cancer in men. The value is normally reported as the number of nanograms of PSA per milliliter of blood. A PSA of \(1.7(\mathrm{ng} / \mathrm{mL})\) is considered low. Express that value in (a) \(\mathrm{g} / \mathrm{L}\), (b) standard SI units of \(\mathrm{kg} / \mathrm{m}^{3}\), and (c) \(\mu \mathrm{g} / \mathrm{L}\).

Complete the following operations using the correct number of significant figures: A. 5.36 × 2.0 = _____ B. 14.2 2 = _____ C. 2 × 3.14159 = _____ D. 4.040 × 5.55 = _____ E. 4.444 × 3.33 = ______ F. 1000 333.3 = _____ G. 2.244 × 88.66 = ______ H. 133 × 2.000 = _____

The United States is about the only country left that uses the units feet, miles, and gallons. You see some car specifications in a magazine that give fuel efficiency as \(7.6\) kilometers per kilogram of fuel. Given that a mile is \(1.609 \mathrm{~km}\), a gallon is \(3.785\) liters, and if a liter of gasoline has a mass of \(0.729 \mathrm{~kg}\), what is the car's fuel efficiency in miles per gallon? SSM

Complete the following operations using the correct number of significant figures: SSM A. 4.55 +21.6 B. 80.00 -112.3 C. 71.1 +3.70 D. 200 +33.7

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