Calculating Work Done A Physics Problem Solved

#main-title

Hey guys! Today, we're diving into a super cool physics problem that involves calculating the work done on an object. We'll be using the work-energy theorem, which is a fundamental concept in mechanics. So, let's break down the problem step by step and make sure we understand every bit of it. We have a body with a mass of 0.15 kg, initially at rest. A force acts upon it, causing it to reach a velocity of 4.0 m/s. Our mission is to find the work done by this force, using the formula W = ΔK, where ΔK represents the change in kinetic energy. Let's get started!

Understanding the Work-Energy Theorem

The work-energy theorem is a cornerstone of classical mechanics, and it's essential to understand it before we jump into calculations. Essentially, this theorem states that the work done on an object is equal to the change in its kinetic energy. Kinetic energy, by the way, is the energy an object possesses due to its motion. Imagine pushing a box across the floor; the work you do on the box increases its kinetic energy, making it move faster. This principle is not just a theoretical concept; it's a practical tool that helps us analyze and solve various physics problems, from simple movements to complex systems. The theorem provides a direct link between force, displacement, and change in speed, allowing us to bypass complicated dynamics calculations in some scenarios. To truly grasp this theorem, think about scenarios you encounter daily. When you kick a ball, you're doing work on it, and the ball gains kinetic energy. Similarly, when brakes are applied in a car, the work done by the brakes reduces the car's kinetic energy, bringing it to a halt. The beauty of the work-energy theorem lies in its simplicity and broad applicability, making it a must-know concept for anyone studying physics. In mathematical terms, work (W) can be expressed as the change in kinetic energy (ΔK), which is the final kinetic energy (Kf) minus the initial kinetic energy (Ki). So, W = ΔK = Kf - Ki. Understanding this formula and the concepts behind it will help you tackle a variety of physics problems with confidence and clarity.

Problem Setup: Identifying Knowns and Unknowns

Alright, let's get into the specifics of our problem. A crucial step in solving any physics problem is to first identify what we know and what we need to find out. This helps us organize our thoughts and determine the best approach. In our scenario, we have a body with a mass (m) of 0.15 kg. This is a key piece of information because mass is a fundamental property that affects an object's inertia and kinetic energy. Next, we know the body is initially at rest. This means its initial velocity (vi) is 0 m/s. This tidbit is super important because it tells us the initial kinetic energy is zero, simplifying our calculations quite a bit. The body then reaches a final velocity (vf) of 4.0 m/s. This is the speed the body attains after the force has acted upon it. Now, what are we trying to find? Our goal is to determine the work (W) done on the body. This is the amount of energy transferred to the body, causing it to change its state of motion. We're given the formula W = ΔK, which links the work done to the change in kinetic energy. So, essentially, we need to calculate the change in kinetic energy to find the work done. This involves calculating both the initial and final kinetic energies. By systematically identifying these knowns and unknowns, we've set a clear path for solving the problem. We know the mass, initial velocity, final velocity, and the formula we need to use. Now it's just a matter of plugging in the values and doing the math!

Calculating Initial Kinetic Energy

The first step in determining the work done is to calculate the initial kinetic energy of the body. Remember, kinetic energy is the energy an object possesses due to its motion, and it's given by the formula: KE = (1/2) * m * v^2, where KE is the kinetic energy, m is the mass, and v is the velocity. In our case, the body is initially at rest, which means its initial velocity (vi) is 0 m/s. This makes our calculation incredibly straightforward. Plugging the values into the formula, we get: Initial KE (Ki) = (1/2) * 0.15 kg * (0 m/s)^2. Now, anything multiplied by zero is zero, so: Ki = (1/2) * 0.15 kg * 0 = 0 Joules. So, the initial kinetic energy of the body is 0 Joules. This makes sense because if an object isn't moving, it doesn't have any kinetic energy. This simplifies our problem significantly because now we only need to calculate the final kinetic energy to find the change in kinetic energy. Understanding that an object at rest has zero kinetic energy is a fundamental concept in physics and helps in solving problems related to motion and energy more efficiently. Knowing the initial kinetic energy is zero gives us a solid foundation to move forward and calculate the final kinetic energy, which will ultimately lead us to finding the work done. So, let's move on to calculating the final kinetic energy!

Determining Final Kinetic Energy

Now that we've nailed down the initial kinetic energy, let's calculate the final kinetic energy of the body. We'll use the same formula we used before: KE = (1/2) * m * v^2. This time, however, we're dealing with the final velocity, which we know is 4.0 m/s. We also know the mass of the body is 0.15 kg. So, let's plug those values into the formula: Final KE (Kf) = (1/2) * 0.15 kg * (4.0 m/s)^2. First, we need to square the final velocity: (4.0 m/s)^2 = 16 m^2/s2. Now, we can substitute this back into our equation: Kf = (1/2) * 0.15 kg * 16 m^2/s2. Next, we multiply 0.15 kg by 16 m^2/s2: 0.15 kg * 16 m^2/s2 = 2.4 kgm^2/s2. Finally, we multiply this by 1/2: Kf = (1/2) * 2.4 kgm^2/s2 = 1.2 Joules. So, the final kinetic energy of the body is 1.2 Joules. This means that after the force acted on the body, it gained 1.2 Joules of energy due to its motion. Understanding how to calculate kinetic energy is essential in physics, as it helps us quantify the energy associated with moving objects. Now that we know both the initial and final kinetic energies, we're just one step away from finding the work done. Let's move on to the final calculation!

Calculating the Work Done

Alright, we're in the home stretch! We've calculated both the initial and final kinetic energies, and now we can finally determine the work done on the body. Remember the work-energy theorem: W = ΔK, where ΔK is the change in kinetic energy. And ΔK is simply the final kinetic energy (Kf) minus the initial kinetic energy (Ki): ΔK = Kf - Ki. We found that the initial kinetic energy (Ki) is 0 Joules, and the final kinetic energy (Kf) is 1.2 Joules. So, let's plug these values into our equation: ΔK = 1.2 Joules - 0 Joules = 1.2 Joules. Now that we know the change in kinetic energy, we can find the work done: W = ΔK = 1.2 Joules. So, the work done on the body is 1.2 Joules. This means that the force applied to the body transferred 1.2 Joules of energy, causing it to accelerate from rest to a velocity of 4.0 m/s. Understanding the work-energy theorem allows us to directly relate the work done on an object to its change in kinetic energy, making it a powerful tool in solving mechanics problems. We've successfully calculated the work done by following a step-by-step approach, from identifying the knowns and unknowns to calculating the initial and final kinetic energies, and finally, applying the work-energy theorem. Great job, guys!

Final Answer and Implications

So, after all our calculations, we've arrived at the final answer: the work done on the 0.15 kg body is 1.2 Joules. This result tells us a lot about the energy transfer in this scenario. The force acting on the body did 1.2 Joules of work, which directly translated into an increase in the body's kinetic energy. This energy is what allowed the body to go from being at rest to moving at 4.0 m/s. This problem beautifully illustrates the work-energy theorem in action. It shows how work, a measure of energy transfer, is directly related to the change in kinetic energy, the energy of motion. The work-energy theorem is not just a formula; it's a principle that helps us understand the fundamental relationship between force, motion, and energy. Understanding this concept is crucial for tackling more complex problems in mechanics, such as those involving potential energy, conservation of energy, and power. Furthermore, it highlights the efficiency of energy transfer in physical systems. In our case, all the work done on the body was converted into kinetic energy, assuming there were no other forces like friction acting against the motion. In real-world scenarios, some energy might be lost to friction or other factors, making the analysis a bit more complex. But for this idealized problem, we see a direct and clear relationship between work and kinetic energy. So, the next time you see an object speeding up or slowing down, remember the work-energy theorem, and you'll have a powerful tool to analyze the energy dynamics at play!

Practical Applications of the Work-Energy Theorem

The work-energy theorem isn't just a theoretical concept confined to textbooks; it has a plethora of practical applications in various fields, making it a cornerstone of engineering and physics. One of the most common applications is in mechanical engineering, where it's used to design machines and systems that involve motion and energy transfer. For example, when designing a car engine, engineers use the work-energy theorem to calculate the work done by the engine's forces and how that work translates into the car's kinetic energy and speed. This helps in optimizing engine performance and fuel efficiency. In civil engineering, the theorem is crucial for analyzing the impact forces in structures. When designing bridges or buildings, engineers need to consider the work done by external forces, like wind or seismic activity, and how these forces affect the structural integrity. By understanding the energy transfer, they can design safer and more resilient structures. Sports science also benefits significantly from the work-energy theorem. Coaches and athletes use it to analyze movements and optimize performance. For instance, understanding the work done by a sprinter's muscles and how it translates into kinetic energy can help improve running techniques and training regimes. Similarly, in projectile motion analysis, the theorem helps in calculating the energy required to launch a projectile and its trajectory. In the realm of aerospace engineering, the work-energy theorem is vital for designing aircraft and spacecraft. Engineers use it to calculate the work done by thrust forces and aerodynamic forces, which is essential for understanding the vehicle's motion and energy requirements. The theorem also plays a key role in safety analysis, such as assessing the impact forces during a crash. Beyond these specific fields, the work-energy theorem is a fundamental tool in any situation involving motion and energy transfer. It allows us to quantitatively analyze how forces do work on objects, changing their kinetic energy, and provides insights into optimizing system performance and ensuring safety. So, whether it's designing a rollercoaster, analyzing a golf swing, or launching a satellite, the work-energy theorem is a principle that underpins much of our understanding and engineering prowess.

#keyword-title: Repair Input Keyword

The original input keyword asks about the value of work done on a 0.15 kg body, initially at rest, that reaches a velocity of 4.0 m/s, using the formula W = ΔK. To make this easier to understand, we can rephrase the question as: "How much work is required to accelerate a 0.15 kg object from rest to a speed of 4.0 m/s?"