AP Physics 2023 Exam Solutions|Q2. PART A ( i ) | Complete Step-by-Step Answers and Explanations"

Study Circus
10 Oct 202306:24
EducationalLearning
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TLDRThis script discusses a physics problem involving a cart sliding down an inclined plane. The cart starts from rest, and the acceleration 'a' is to be determined. The script outlines the method to find acceleration by plotting the position 'x' against 'half t²', where 't' is time. The slope of the resulting straight line graph through the origin will represent the acceleration of the cart, as derived from the equation of motion. The explanation provides a clear step-by-step approach to solving the problem using the given data.

Takeaways
  • 📚 The context is a physics problem involving a cart sliding down an inclined plane with the positive x-direction defined along the incline.
  • 🔍 The cart's acceleration, denoted as 'a', is to be determined using the given data and kinematic equations.
  • 🚀 The cart starts from rest, which means the initial velocity (u) is zero, simplifying the displacement equation to "s = 0.5 * a * t^2".
  • 📈 The goal is to create a graph that yields a straight line, the slope of which can be used to determine the acceleration 'a'.
  • 📝 The student is instructed to use the remaining columns in a table to record necessary values for graphing.
  • 📊 For a straight-line graph, the independent variable is chosen as "0.5 * t^2" and the dependent variable as the position 'x'.
  • 📐 The graph's slope will represent the acceleration 'a', which is the key quantity to be determined.
  • 📉 The values of "0.5 * t^2" for given times are calculated and tabulated to be used on the x-axis of the graph.
  • 📝 The position 'x', corresponding to each time, will be plotted on the y-axis to create the straight-line graph.
  • 🔑 The straight-line graph passing through the origin with a slope equal to the acceleration 'a' will be the result of plotting position versus "0.5 * t^2".
  • 🎯 The final takeaway is that plotting position against "0.5 * t^2" will yield the desired straight line, the slope of which is the acceleration of the cart.
Q & A

  • What is the initial condition of the cart in the problem?

    -The cart is released from rest at the top of the ramp, meaning its initial velocity (U) is zero.

  • In which direction is the positive X-axis defined?

    -The positive X-axis is defined along the incline, down the incline.

  • What equation of motion is used to describe the cart's displacement?

    -The equation of motion used is ( s = ut + 1/2at^2 ), where ( s ) is the displacement, ( u ) is the initial velocity, ( a ) is the acceleration, and ( t ) is the time.

  • Why is the initial velocity (U) considered zero in this problem?

    -The initial velocity (U) is zero because the cart is starting from rest.

  • How is the displacement (s) related to the position (x) in this scenario?

    -Since the cart starts from ( x = 0 ), the displacement (s) is equal to the position (x) of the cart.

  • What form of the equation is sought to graph a straight line to determine acceleration?

    -The form of the equation sought is ( y = mx ), where ( y ) is the dependent variable, ( x ) is the independent variable, and ( m ) is the slope of the line.

  • Which quantities should be graphed to yield a straight line for determining the acceleration?

    -The position (x) should be graphed on the y-axis, and half of the square of the time ( 1/2t^2 ) should be graphed on the x-axis.

  • What will be the slope of the straight line graph in this context?

    -The slope of the straight line graph will be the acceleration (a) of the cart.

  • What values need to be calculated to plot the graph?

    -The values of ( 1/2t^2 ) need to be calculated for each given time.

  • How does the graph of position (x) versus half of the square of the time ( 1/2t^2 ) help in determining the acceleration?

    -Plotting the position (x) on the y-axis and ( 1/2t^2 ) on the x-axis will yield a straight line graph passing through the origin, with the slope of the line representing the acceleration (a).

Outlines
00:00
📚 Determining Cart's Acceleration with Graphs

This paragraph discusses a physics problem involving a cart sliding down an inclined plane. The positive X-direction is defined along the incline, and the acceleration 'a' of the cart is to be determined from collected data. The initial velocity is zero, and the displacement 's' of the cart is related to time 't' and acceleration 'a' by the equation s = 1/2 * a * t². The paragraph explains that to find the acceleration, one should plot the position 'x' against half the square of time (1/2 * t²), which will yield a straight line graph with the slope equal to the acceleration 'a'. The student is instructed to use this method to determine the acceleration from the provided data.

05:01
📈 Graphing Half Time Squared to Find Acceleration

The second paragraph continues the explanation of how to use graphical methods to determine the acceleration of the cart. It emphasizes plotting half the square of time (1/2 * t²) on the x-axis against the position 'x' on the y-axis. This will result in a straight line graph that passes through the origin, with the slope of the line representing the acceleration of the cart. The paragraph provides an example of how to calculate the values of 1/2 * t² for different times and suggests tabulating these values for plotting. The summary concludes with the understanding that the slope of the resulting graph will directly give the acceleration of the cart.

Mindmap
Keywords
💡Acceleration
Acceleration refers to the rate at which the velocity of the cart changes as it slides down the inclined plane. In the script, acceleration 'a' is the quantity to be determined from the slope of the straight line graph plotted using the given data. The acceleration is a key concept because it describes how quickly the cart speeds up as it moves down the incline.
💡Inclined Plane
An inclined plane is a flat surface tilted at an angle, used in this context as the ramp down which the cart slides. The inclined plane provides a practical scenario to apply principles of physics, such as motion and acceleration, as the cart's movement along the plane is influenced by gravity.
💡Displacement
Displacement, denoted as 's', is the distance moved by the cart from its starting position. In the script, displacement is a crucial measure because it helps to determine the position of the cart at various time intervals, which is then used to calculate acceleration.
💡Equation of Motion
The equation of motion, s = ut + (1/2)at², is used to describe the displacement 's' of an object moving under constant acceleration. In the video, this equation is simplified to s = (1/2)at² since the cart starts from rest (initial velocity 'u' is zero). This relationship is used to find the acceleration by plotting the displacement against time squared.
💡Graph
A graph is a visual representation of data. In the script, the student needs to plot a graph of displacement 'x' versus half the square of time '(1/2)t²'. This graph helps to determine the acceleration 'a' as the slope of the line, highlighting the importance of graphical analysis in physics.
💡Slope
The slope of a graph represents the rate of change of the dependent variable with respect to the independent variable. In this context, the slope of the displacement versus half time squared graph corresponds to the acceleration 'a' of the cart. Identifying the slope is critical for calculating the acceleration.
💡Initial Velocity
Initial velocity, denoted as 'u', is the velocity of the cart at the starting point. Since the cart is released from rest, its initial velocity is zero, simplifying the equation of motion used to calculate displacement. This condition is important for applying the correct form of the equation.
💡Straight Line
A straight line graph indicates a linear relationship between the variables. In the video, plotting displacement against half time squared should yield a straight line passing through the origin if the cart's motion is uniformly accelerated. The straight line's slope directly relates to the cart's acceleration.
💡Position
Position refers to the location of the cart along the inclined plane at different times. The position data provided in the table is used to create the graph that will ultimately determine the cart's acceleration. Position is essential for tracking the motion of the cart.
💡Independent Variable
The independent variable is the variable that is manipulated or controlled in an experiment. In the script, half the square of time '(1/2)t²' is treated as the independent variable when plotting the graph. This choice facilitates a straight-line relationship, allowing the calculation of acceleration from the slope.
Highlights

The second question of the 2023 free response questions involves a cart sliding down an inclined plane with the positive X direction along the incline.

The cart starts from rest at the top of the ramp, with acceleration denoted as 'a'.

Displacement of the cart is given by the equation s = ut + (1/2)at², with initial velocity u being zero.

To find acceleration, a straight line graph is needed, where the slope represents acceleration.

The graph should have the form y = mx, where m is the slope and the graph passes through the origin.

For the given situation, x is taken as (1/2)at² and treated as the independent variable.

The dependent variable is the position X, which is plotted on the y-axis.

Plotting (1/2)at² on the x-axis and position X on the y-axis will yield a straight line graph.

The slope of this graph is equal to the acceleration of the cart.

Values of (1/2)t² are calculated for the given times to use as x-axis values.

A table of time and position data is provided to determine the acceleration.

The acceleration can be determined from the slope of the straight line graph passing through the origin.

The method involves using the equation of motion and graphing the relationship between displacement and time.

The problem demonstrates the application of kinematic equations to determine acceleration from a graph.

The approach highlights the importance of understanding the relationship between variables in kinematics.

The solution process involves identifying the correct variables to plot and understanding their significance in determining acceleration.

The problem tests the ability to apply mathematical concepts to physical situations, specifically in the context of motion.

The method provides a clear and logical step-by-step approach to solving the problem of finding acceleration.

Transcripts

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Related Tags
PhysicsMotionAccelerationInclined PlaneCartEquation of MotionSlopeGraph AnalysisEducationalScience