'AP Physics 2023 Exam Solutions|Q1. PART C ( ii ) | Complete Step-by-Step Answers and Explanations"

Study Circus
6 Oct 202304:50
EducationalLearning
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TLDRThe video script discusses a physics problem involving a block dropped onto a cart attached to a spring. The total energy of the system is 4 joules, which is conserved and divided between the kinetic energy of the cart and the potential energy of the spring. The task is to plot the kinetic energy of the cart against the potential energy, resulting in a linear graph with a slope of -1/4 and an intercept of 1. This graph illustrates the energy transfer between the cart's motion and the spring's deformation over time.

Takeaways
  • 📉 The task involves plotting a graph of kinetic energy versus potential energy for a system consisting of a cart and a spring, excluding the block.
  • 🔍 The total energy of the system is given as 4 joules.
  • ⚙️ The total energy is the sum of the kinetic energy and the potential energy of the system.
  • 🧩 The kinetic energy of the system includes the kinetic energy of both the cart and the block.
  • ✏️ The equation for the total energy is 4 = 0.5mv² + 0.5kx².
  • 📝 The goal is to plot kinetic energy (Y variable) against potential energy (X variable).
  • 📊 The derived equation for the graph is Y = -X/4 + 1.
  • 📐 The slope of the graph is -1/4.
  • 🟢 The Y-intercept of the graph is 1.
  • 🔴 The X-intercept of the graph is 4, giving a straight line graph with the specified slope and intercepts.
Q & A

  • What is the total energy of the system consisting of the cart and the spring?

    -The total energy of the system is 4 joules, which is the sum of kinetic and potential energy at any given position.

  • What is the formula for the potential energy of the spring in the system?

    -The potential energy of the spring is given by \( \frac{1}{2} K x^2 \), where \( K \) is the spring constant and \( x \) is the displacement from the equilibrium position.

  • How is the kinetic energy of the cart represented in the system?

    -The kinetic energy of the cart is represented as \( \frac{1}{2} M v^2 \), where \( M \) is the mass of the cart and \( v \) is its velocity.

  • What is the relationship between the kinetic energy of the cart and the total energy of the system?

    -The kinetic energy of the cart is part of the total energy of the system. The total energy is the sum of the kinetic energy of the cart and the potential energy of the spring.

  • What is the equation that represents the graph of kinetic energy of the cart versus the potential energy?

    -The equation representing the graph is \( y = -\frac{x}{4} + 1 \), where \( y \) is the kinetic energy of the cart and \( x \) is the potential energy of the system.

  • What is the slope of the graph representing the kinetic energy of the cart?

    -The slope of the graph is \( -\frac{1}{4} \), indicating the rate of change of kinetic energy with respect to potential energy.

  • What is the y-intercept of the graph for the kinetic energy of the cart?

    -The y-intercept of the graph is 1, which represents the kinetic energy of the cart when the potential energy is zero.

  • What does the x-intercept of the graph indicate?

    -The x-intercept, where \( y = 0 \), indicates the potential energy when the kinetic energy of the cart is zero, which is 4 joules in this case.

  • How can you determine two points on the graph to draw a straight line?

    -By using the y-intercept and the x-intercept, or by knowing the total energy and the conditions when the cart is at rest or at maximum displacement.

  • What does the red line in the graph represent?

    -The red line represents the relationship between the kinetic energy of the cart and the potential energy of the system at any given time, illustrating the conservation of energy.

  • Why is the graph important for understanding the second part of the question?

    -The graph is important because it visually demonstrates the conversion between kinetic and potential energy in the system, which is key to solving the problem presented in the second part of the question.

Outlines
00:00
📚 Kinetic and Potential Energy Graphing Task

The paragraph explains a physics problem involving a block dropped onto a cart with a spring. The task is to sketch a graph representing the kinetic energy of the system consisting only of the cart and the spring, excluding the block. The total energy of the system is given as 4 joules, which is the sum of kinetic and potential energy. The kinetic energy of the cart and block are detailed, and the relationship between total energy, kinetic energy of the cart (Y variable), and potential energy (X variable) is established. The goal is to plot a graph with a specific equation that relates Y to X, showcasing the energy transformation within the system.

Mindmap
Keywords
💡Kinetic Energy
Kinetic energy is the energy possessed by an object due to its motion. In the context of the video, the kinetic energy of the cart is a central concept, as it is part of the system's total energy. The script discusses how the kinetic energy of the cart and the block combine to form the system's total energy, which is 4 joules. The kinetic energy of the cart is represented as 'Y' in the graph, and it is shown to be inversely proportional to the potential energy 'X', illustrating the conservation of energy in the system.
💡Potential Energy
Potential energy is the stored energy of an object due to its position in a force field, such as a spring. In the video, potential energy is discussed in relation to the spring's energy when the block is dropped onto the cart. The script mentions that the potential energy is represented as 'X' in the graph and is part of the total energy of the system. It is shown to decrease as the kinetic energy of the cart increases, demonstrating the transformation between potential and kinetic energy.
💡System
In the video, the term 'system' refers to the collection of objects being studied, which includes the cart and the spring. The script emphasizes that the total energy of the system is conserved, meaning it remains constant throughout the motion. The system's energy is the sum of the kinetic and potential energies, and the script uses the system to illustrate the principles of energy conservation and transformation.
💡Conservation of Energy
The conservation of energy is a fundamental principle stating that energy cannot be created or destroyed, only transformed from one form to another. The script uses the example of the block dropping onto the cart to explain this principle, showing that the total energy of the system remains 4 joules, regardless of the distribution between kinetic and potential energy.
💡Cart
The cart in the video is one of the components of the system being analyzed. It has mass and can move, thus possessing kinetic energy. The script describes the cart's kinetic energy as a variable 'Y', which is plotted against the potential energy 'X' to demonstrate the relationship between the two energies in the system.
💡Block
The block is another component of the system that interacts with the cart when it is dropped onto it. The script mentions the block's kinetic energy as it contributes to the total energy of the system. The block's impact on the cart causes a transfer of energy, which is a key part of the energy conservation concept discussed in the video.
💡Graph
A graph is a visual representation of data, used in the video to illustrate the relationship between the kinetic and potential energies. The script describes the process of plotting a graph with kinetic energy on the Y-axis and potential energy on the X-axis, showing how these energies are inversely related and sum up to the total energy of the system.
💡Total Energy
Total energy refers to the sum of all forms of energy in a system. In the script, the total energy is given as 4 joules and is depicted as a constant value, emphasizing the principle of energy conservation. The total energy is the sum of the kinetic energy of the cart and the block, and the potential energy stored in the spring.
💡Mass
Mass is a measure of the amount of matter in an object and plays a crucial role in calculating kinetic energy. The script mentions 'm' as the mass of the cart and '3m' as the combined mass of the cart and the block, which are used in the formulas for kinetic energy to show how mass affects the energy of the system.
💡Velocity
Velocity is the speed of an object in a particular direction. In the video, velocity is implied through the discussion of kinetic energy, which is calculated as half the mass times the velocity squared. The script does not explicitly mention velocity, but it is understood to be a key factor in determining the kinetic energy of the cart.
💡Spring
The spring in the video is a mechanical device that stores potential energy when compressed or stretched. The script describes the spring's role in the system, where its potential energy changes as the block is dropped onto the cart, leading to a conversion of energy from the spring to the cart's kinetic energy.
Highlights

The task involves analyzing the kinetic energy of a system consisting of a cart and a spring after a block is dropped onto the cart.

The system's total energy is conserved and is given as 4 joules, distributed between kinetic and potential energy.

The potential energy is represented by the spring's energy, calculated as \( \frac{1}{2} K x^2 \).

The kinetic energy of the system is the sum of the kinetic energies of the cart and the block.

The kinetic energy of the cart is represented as \( \frac{1}{2} M v^2 \), where M is the mass of the cart and v is its velocity.

The kinetic energy of the block is represented as \( \frac{1}{2} m v^2 \), with m being the mass of the block.

The total energy of the system is the sum of the kinetic and potential energies, which must equal 4 joules at any position.

The task requires plotting the kinetic energy of the cart (Y variable) against the potential energy (X variable).

The equation for the graph is derived as \( 4y = 4 - x \), simplifying to \( y = -\frac{x}{4} + 1 \).

The graph has a slope of -1/4, indicating the relationship between the kinetic and potential energies.

The Y-intercept of the graph is 1, representing the maximum kinetic energy of the cart when the potential energy is zero.

The X-intercept occurs when Y equals zero, indicating the potential energy is 4 joules, all of which is stored as potential energy.

The graph represents the kinetic energy of the cart versus the potential energy of the system at any given time.

The red line in the graph symbolizes the conservation of energy within the system.

The explanation provides a clear understanding of how to plot and interpret the energy graph for the given system.

The second part of the question requires finding and explaining the graph of the cart's kinetic energy in relation to the system's potential energy.

The explanation concludes with a sufficient understanding of the task and its practical implications in energy conservation.

Transcripts

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Kinetic EnergyPotential EnergyCart DynamicsSpring SystemEnergy ConservationPhysics ProblemGraphical AnalysisMechanical EnergyBlock InteractionEnergy Graph