Calculating Final Velocity Of A Cart On An Inclined Plane

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Calculating Final Velocity of a Cart on an Inclined Plane

Hey guys! Let's dive into a classic physics problem: figuring out the final speed of a cart rolling down a frictionless incline. We've got a little 8 kg cart, it's starting with an initial velocity, and it's going to slide down a slope. The question asks to calculate the final velocity of the cart if the height of the incline is 20 m and the angle of elevation is 30°. This is a great example of how to apply the principles of energy conservation. Get ready to flex those physics muscles! Understanding this type of problem is crucial for grasping concepts in mechanics. It's not just about getting an answer; it's about understanding the 'why' behind it. This problem requires us to consider the initial energy of the cart (kinetic energy due to its initial velocity and potential energy due to its height), and how that energy transforms as the cart rolls down the incline. By understanding these concepts, you'll be well-equipped to tackle more complex physics problems. Let's start with the basics.

Understanding the Problem and Key Concepts

Okay, so what are we dealing with? We have a cart (8 kg) that's initially moving (7 m/s) and is at a certain height (which we'll figure out from the 20 m height and 30° angle). The key thing is that the incline is frictionless. This means we don't have to worry about energy loss due to friction. We are dealing with an example of mechanical energy conservation. This is a fundamental principle in physics that states the total mechanical energy of a system remains constant if only conservative forces are doing work. In our case, the only force doing work is gravity, which is a conservative force. The cart will convert some of its potential energy into kinetic energy as it descends.

Before we jump into the math, let's break down the types of energy involved. First, we have kinetic energy, which is the energy of motion. The faster the cart moves, the more kinetic energy it has. The cart also has potential energy. Since the cart is at a certain height above the ground, it possesses gravitational potential energy. This is energy stored due to its position in a gravitational field. As the cart rolls down, its potential energy decreases as it loses height, and its kinetic energy increases as it gains speed. To solve this problem, we'll need a few key concepts and formulas:

  • Kinetic Energy (KE): KE = 1/2 * m * v^2, where 'm' is the mass and 'v' is the velocity.
  • Potential Energy (PE): PE = m * g * h, where 'm' is the mass, 'g' is the acceleration due to gravity (approximately 9.8 m/s²), and 'h' is the height.
  • Conservation of Mechanical Energy: Initial KE + Initial PE = Final KE + Final PE.

By understanding these formulas and the principle of conservation of energy, we can easily solve for the cart's final velocity. This understanding is useful not just in physics classes, but also in many real-world scenarios. For example, understanding energy conservation is important in designing roller coasters, analyzing the motion of vehicles, and even in understanding the flow of energy in ecosystems. So let's get into the calculation phase, it will be fun, I promise! The calculations will show you exactly how to approach this type of problem.

Setting Up the Problem and Calculating Initial Energy

Alright, let's get down to business. We've got our cart, and we need to figure out its initial energy state. First, the cart has both kinetic and potential energy at the start. So, we'll calculate them step by step. Let's identify the variables we know and those we need to find. We are given the mass of the cart (m = 8 kg), the initial velocity (vi = 7 m/s), the height of the incline (h = 20 m) and the angle of inclination (θ = 30°). We need to determine the final velocity (vf). First let's calculate the initial kinetic energy (KEi): KEi = 1/2 * m * vi^2 = 1/2 * 8 kg * (7 m/s)^2. KEi = 196 Joules. The initial potential energy (PEi) is based on the height, but we need to calculate the initial height of the cart to apply the PE formula. If we have the height and the angle of inclination, we can infer the length of the incline, and the height that the cart is at. Now, to find the initial height of the cart, we can use the following formula h = L * sin(θ), where L is the length of the incline, and θ is the angle. It means that the vertical height of the cart does not depend on the incline angle. So at the beginning the cart is at a height of 20 m. So, PEi = m * g * h = 8 kg * 9.8 m/s² * 20 m = 1568 Joules.

So, the total initial energy (TEi) is the sum of KEi and PEi: TEi = KEi + PEi. TEi = 196 J + 1568 J = 1764 J. This total energy represents the total mechanical energy the cart has at the beginning of its journey down the incline. This is the energy that will be conserved throughout the cart's motion, meaning it will be transformed between kinetic and potential energy. The beauty of this is that the total amount remains the same, assuming no energy is lost to friction or other non-conservative forces. The cart is going to start rolling, and it will pick up speed as it goes down the incline. As it loses potential energy, it gains kinetic energy. At the bottom of the incline, all the potential energy is converted to kinetic energy. Let's find out how! To do this, we'll use the principle of conservation of energy, which states that the total mechanical energy of the system remains constant (assuming no energy is lost to friction or other non-conservative forces). This means the total energy at the beginning (TEi) equals the total energy at the end (TEf).

Applying Conservation of Energy to Find Final Velocity

Okay, we've got the initial energy. Now, let's figure out what's happening at the end. At the bottom of the incline, the cart's potential energy will be zero (since we'll consider the bottom as the zero-potential-energy reference point). All of the initial potential energy will have been converted to kinetic energy. The total final energy (TEf) will only be kinetic energy. So, we know that TEi = TEf. We can write this as KEi + PEi = KEf + PEf. Let's plug in what we know: 196 J + 1568 J = KEf + 0. Since TEi = TEf, we also know that 1764 J = KEf. And since KEf = 1/2 * m * vf^2, we can rearrange the formula to solve for the final velocity (vf). Let's plug in the known values: 1764 J = 1/2 * 8 kg * vf^2. Simplify: 1764 J = 4 kg * vf^2. Divide both sides by 4 kg: vf^2 = 441 m²/s². Take the square root of both sides: vf = √441 m²/s². Therefore, vf = 21 m/s.

So, the final velocity of the cart is 21 m/s. This makes sense because the cart is moving faster at the end. The cart started with a velocity of 7 m/s, and it gained speed as it went down the hill. This increase in speed is due to the conversion of potential energy to kinetic energy. Because the incline is frictionless, there's no energy loss, and all the initial energy is conserved throughout the movement of the cart. This problem perfectly illustrates the conservation of mechanical energy. It shows how potential energy, due to the cart's initial height, is converted into kinetic energy, increasing its velocity as it rolls down the incline. By understanding these principles, you can solve similar problems involving energy transformations in various physical systems, such as roller coasters or pendulums.

Summary and Key Takeaways

Alright, let's recap what we've learned and what the key takeaways are from this problem. We successfully calculated the final velocity of an 8 kg cart rolling down a 20 m high frictionless incline with an initial velocity of 7 m/s and an incline angle of 30°. We did this by applying the principle of conservation of mechanical energy. We used the formulas for kinetic energy (KE = 1/2 * m * v^2) and potential energy (PE = m * g * h). We calculated the initial total energy (kinetic + potential), and then we used the conservation principle (total initial energy = total final energy) to solve for the final velocity. We found the final velocity to be 21 m/s. This is a great example of how energy transforms from one form to another (potential to kinetic) in a closed system. The key takeaways from this problem are:

  • Understanding Energy Conservation: The total mechanical energy (kinetic + potential) of a system remains constant if no external forces (like friction) do work.
  • Energy Transformation: Potential energy can convert into kinetic energy, resulting in an increase in velocity.
  • Problem-Solving Approach: Break down the problem into initial and final states, identify the relevant energy forms, and apply the conservation principle.

These concepts are fundamental to understanding physics and are applicable in many real-world scenarios. I hope this was helpful, guys! Keep practicing, and you'll get the hang of it! Understanding the concepts of potential and kinetic energy is fundamental to understanding physics, from the motion of a ball to the design of a roller coaster. The more you practice these types of problems, the better you'll become at applying these concepts and solving more complex problems. Keep up the great work, and don't be afraid to ask questions! The principles of physics are beautiful and powerful, and they apply to everything from the smallest particles to the largest galaxies. Keep exploring the world around you and enjoy the journey of learning and discovery!