Textbook Question
Sketch the v vs. t graph for the object whose displacement as a function of time is given by Fig. 2–40.
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2. 1D Motion / Kinematics
Graphing Position, Velocity, and Acceleration Graphs
5:29 minutesProblem 5a
Textbook Question
FIGURE EX2.5 shows the position graph of a particle. Draw the particle’s velocity graph for the interval .
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Analyze the position vs. time graph provided in the problem. Identify the key features such as slopes (positive, negative, or zero) and any changes in the slope, as these correspond to the velocity of the particle.
Recall that the velocity of a particle is the derivative of its position with respect to time, \( v(t) = \frac{dx}{dt} \). The slope of the position graph at any point gives the velocity at that time.
Break the position graph into segments where the slope is constant or changes. For each segment, determine whether the slope is positive (indicating positive velocity), negative (indicating negative velocity), or zero (indicating zero velocity).
Plot the velocity graph based on the slopes of the position graph. For example, if the slope of the position graph is constant and positive, the velocity graph will show a horizontal line at a positive value. If the slope is zero, the velocity graph will show a horizontal line at zero.
Ensure that the transitions in the velocity graph correspond to the changes in slope of the position graph. For instance, if the slope of the position graph changes abruptly, the velocity graph will show a discontinuity at that point.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Position vs. Time Graphs
A position vs. time graph illustrates how the position of a particle changes over time. The slope of this graph at any point represents the velocity of the particle. A steeper slope indicates a higher velocity, while a flat slope indicates the particle is at rest. Understanding this relationship is crucial for interpreting how the particle moves.
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Velocity
Velocity is a vector quantity that describes the rate of change of an object's position with respect to time. It has both magnitude and direction, distinguishing it from speed, which is scalar. The velocity graph can be derived from the position graph by calculating the slope of the position curve at various points, indicating how quickly and in which direction the particle is moving.
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Graphical Interpretation
Graphical interpretation involves analyzing graphs to extract meaningful information about physical phenomena. In this context, converting a position graph to a velocity graph requires understanding how to represent changes in position as changes in velocity over time. This skill is essential for visualizing motion and predicting future behavior of the particle based on its current state.
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