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

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

- π 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.
- π The total energy of the system is given as 4 joules.
- βοΈ The total energy is the sum of the kinetic energy and the potential energy of the system.
- π§© The kinetic energy of the system includes the kinetic energy of both the cart and the block.
- βοΈ The equation for the total energy is 4 = 0.5mvΒ² + 0.5kxΒ².
- π The goal is to plot kinetic energy (Y variable) against potential energy (X variable).
- π The derived equation for the graph is Y = -X/4 + 1.
- π The slope of the graph is -1/4.
- π’ The Y-intercept of the graph is 1.
- π΄ 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

##### π 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

##### π‘Kinetic Energy

##### π‘Potential Energy

##### π‘System

##### π‘Conservation of Energy

##### π‘Cart

##### π‘Block

##### π‘Graph

##### π‘Total Energy

##### π‘Mass

##### π‘Velocity

##### π‘Spring

###### 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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