Work Done Calculation Runner's Kinetic Energy Change

In the realm of physics, the relationship between work and energy is fundamental to understanding motion and its changes. The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. This principle provides a powerful tool for analyzing scenarios where forces cause changes in an object's speed. In this article, we will explore a specific example involving a runner who accelerates, and we will calculate the work done using the work-energy principle. This exploration will not only solidify your understanding of the principle but also enhance your problem-solving skills in physics.

The work-energy principle is a cornerstone of classical mechanics, linking the concepts of work and energy in a clear and concise manner. It's essential to grasp this principle to analyze various physical situations, such as the motion of objects under the influence of forces. Understanding how work translates into changes in kinetic energy allows us to predict and explain the behavior of objects in motion. In the context of this article, we will apply this principle to determine the amount of work done on a runner as they increase their speed. The detailed calculation and explanation will provide a comprehensive understanding of the relationship between work, energy, and motion.

Kinetic energy, defined as the energy an object possesses due to its motion, plays a crucial role in this principle. It depends on both the object's mass and its velocity, as described by the equation $KE = (1/2)mv^2$, where KE represents kinetic energy, m is the mass, and v is the velocity. Changes in kinetic energy directly reflect the work done on the object. The higher the change in velocity, the greater the change in kinetic energy, and thus the more work done. This relationship is vital in many real-world applications, from understanding the motion of vehicles to the dynamics of sports activities. By examining how kinetic energy changes in different scenarios, we can better understand the forces at play and the work being done. This article will demonstrate how to calculate the change in kinetic energy and relate it to the work done on the runner, providing a practical application of the theoretical concepts.

Let’s consider a scenario where a runner with a mass of 60.0 kg is initially moving at a speed of 6.00 m/s. The runner then accelerates, increasing their speed to 9.00 m/s. Our objective is to determine the amount of work done during this acceleration. This problem is a classic example of applying the work-energy principle, where we can relate the change in the runner's kinetic energy to the work done on them. This scenario helps illustrate how forces exerted by the runner (such as through their muscles) translate into changes in motion and energy. By solving this problem, we will reinforce the understanding of key concepts in mechanics, such as kinetic energy, work, and their interrelation.

To solve this problem effectively, we need to carefully analyze the given information and apply the relevant physics principles. The initial conditions, including the runner's mass and initial velocity, are crucial for calculating the initial kinetic energy. Similarly, the final velocity is necessary to determine the final kinetic energy. The change in kinetic energy, which is the difference between the final and initial kinetic energies, will then allow us to find the work done. This methodical approach is essential for solving physics problems accurately. It ensures that all the necessary variables are considered and that the calculations are performed correctly. In the following sections, we will break down the steps to solve this problem, providing a clear and detailed explanation of the process.

Additionally, understanding the context of the problem is important. In this case, the work done on the runner is primarily due to the forces exerted by the runner themselves, pushing against the ground. These internal forces result in an external force that propels the runner forward, increasing their speed and kinetic energy. It is also worth noting that we are assuming no other significant forces are acting on the runner, such as air resistance or friction, for the sake of simplicity. This assumption allows us to focus solely on the work-energy relationship and provides a clear example of how it works in an idealized scenario. The next sections will delve into the calculations needed to solve the problem, making the connection between the runner's motion and the work done more explicit.

To calculate the work done on the runner, we will follow these steps:

1. Calculate the initial kinetic energy (KEᵢ): The formula for kinetic energy is KE = (1/2)mv², where m is the mass and v is the velocity. KEᵢ = (1/2) * 60.0 kg * (6.00 m/s)² KEᵢ = (1/2) * 60.0 kg * 36.0 m²/s² KEᵢ = 30.0 kg * 36.0 m²/s² KEᵢ = 1,080 J

2. Calculate the final kinetic energy (KEf): Using the same formula: KEf = (1/2) * 60.0 kg * (9.00 m/s)² KEf = (1/2) * 60.0 kg * 81.0 m²/s² KEf = 30.0 kg * 81.0 m²/s² KEf = 2,430 J

3. Calculate the change in kinetic energy (ΔKE): The change in kinetic energy is the final kinetic energy minus the initial kinetic energy. ΔKE = KEf - KEᵢ ΔKE = 2,430 J - 1,080 J ΔKE = 1,350 J

4. Determine the work done (W): According to the work-energy principle, the work done is equal to the change in kinetic energy. W = ΔKE W = 1,350 J

Thus, the work done on the runner as they accelerated from 6.00 m/s to 9.00 m/s is 1,350 J. This positive value indicates that work was done on the runner, increasing their kinetic energy. The detailed step-by-step calculation provided above ensures clarity and accuracy in understanding how the work-energy principle is applied in this scenario. This methodical approach is crucial for solving physics problems, as it helps break down complex situations into manageable steps. In the following section, we will discuss the implications of this result and further explore the relationship between work and energy.

Furthermore, the calculations demonstrate the direct relationship between the square of the velocity and kinetic energy. As the runner's velocity increased from 6.00 m/s to 9.00 m/s, the kinetic energy more than doubled, which is reflected in the work done. This illustrates the significant impact of velocity on an object's kinetic energy. It's also important to note that the mass of the runner plays a crucial role in determining the kinetic energy. A heavier runner would require more work to achieve the same change in velocity. The work-energy principle provides a quantitative way to understand these relationships, allowing us to predict the energy changes associated with motion. By understanding these principles, we can analyze and interpret various physical phenomena involving work and energy.

Based on our calculations, the work done on the runner as they accelerated from 6.00 m/s to 9.00 m/s is 1,350 J. Therefore, the correct answer is C. 1,350 J.

In conclusion, we have successfully calculated the work done on a runner accelerating from 6.00 m/s to 9.00 m/s using the work-energy principle. The work done, 1,350 J, corresponds directly to the change in the runner's kinetic energy. This problem effectively illustrates the application of a fundamental concept in physics, highlighting the relationship between work and energy. The work-energy principle is a powerful tool for analyzing motion and energy transformations in various physical systems. Understanding this principle allows us to predict and explain how forces cause changes in an object's kinetic energy. It simplifies the analysis of motion by directly linking the work done to the change in kinetic energy, bypassing the need to consider the details of the forces involved over time.

Moreover, the application of this principle extends beyond simple scenarios like this runner's acceleration. It is a core concept in more complex systems, such as machines, vehicles, and even biological systems. For instance, engineers use the work-energy principle to design efficient engines and machines by optimizing the transfer of energy. In biomechanics, it helps in understanding human movement and the energy requirements for various activities. The principle's broad applicability underscores its importance in both theoretical physics and practical applications. By mastering the work-energy principle, one gains a deeper understanding of the physical world and the underlying mechanisms that govern motion and energy.

In summary, the problem we solved demonstrates the practical application of the work-energy principle in a real-world scenario. By calculating the change in kinetic energy and equating it to the work done, we have shown how this principle can be used to quantify the energy transformations involved in motion. The detailed calculations and explanations provided in this article offer a comprehensive understanding of the principle and its significance in physics. This knowledge serves as a foundation for tackling more complex problems and further exploring the fascinating world of mechanics and energy.