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x(t) (Position) Time (s) v(t) (Velocity) Time (s)

x(t) (Position) Time (s) v(t) (Velocity) Time (s)

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Sketch the velocity vs time graph from the given position vs time graph
Welcome to our educational resource on motion dynamics. Below are detailed explanations and transcriptions of two critical graphs used to illustrate concepts of position and velocity over time. ### Graph 1: Position vs. Time This graph plots the position \( x(t) \) of an object over time \( t \). **Axes:** - The horizontal axis represents time (t) in seconds (s). - The vertical axis represents position \( x(t) \) in meters. **Graph Description:** - The curve begins from a positive position at \( t = 0 \), initially moving upward. - As time progresses, the position increases to a peak, then returns to a neutral position (x = 0). - After crossing the neutral position, the curve dips into negative territory, reaching a low point. - Finally, the curve ascends back into positive values by the end of the observed time frame. This sinusoidal-like curve indicates oscillatory motion, possibly due to harmonic oscillation or periodic force. ### Graph 2: Velocity vs. Time This graph represents the velocity \( v(t) \) of the same object over time \( t \). **Axes:** - The horizontal axis again represents time (t) in seconds (s). - The vertical axis represents velocity \( v(t) \) in meters per second (m/s). **Graph Description:** - This is a blank graph and does not display any velocity data. - The lack of a plotted curve suggests that this graph either needs to be filled with corresponding velocity data or it could be an exercise for students to practice plotting velocity based on position data from the first graph. ### Interpretation and Relationships: Understanding the relationship between these two graphs is integral: 1. **Position to Velocity:** The velocity graph is the derivative of the position graph. For instance, when the position curve peaks (indicating maximum or minimum), the velocity curve would cross zero because the rate of change at these points is zero. 2. **Velocity to Position:** Conversely, the position graph can be derived by integrating the velocity graph. When velocity is zero, it indicates that the object is at an extreme (either maximum or minimum) or changing direction. This graphical representation helps in visualizing how an object’s motion changes over time and forms the basis of kinematic analysis in physics.
Transcribed Image Text:Welcome to our educational resource on motion dynamics. Below are detailed explanations and transcriptions of two critical graphs used to illustrate concepts of position and velocity over time. ### Graph 1: Position vs. Time This graph plots the position \( x(t) \) of an object over time \( t \). **Axes:** - The horizontal axis represents time (t) in seconds (s). - The vertical axis represents position \( x(t) \) in meters. **Graph Description:** - The curve begins from a positive position at \( t = 0 \), initially moving upward. - As time progresses, the position increases to a peak, then returns to a neutral position (x = 0). - After crossing the neutral position, the curve dips into negative territory, reaching a low point. - Finally, the curve ascends back into positive values by the end of the observed time frame. This sinusoidal-like curve indicates oscillatory motion, possibly due to harmonic oscillation or periodic force. ### Graph 2: Velocity vs. Time This graph represents the velocity \( v(t) \) of the same object over time \( t \). **Axes:** - The horizontal axis again represents time (t) in seconds (s). - The vertical axis represents velocity \( v(t) \) in meters per second (m/s). **Graph Description:** - This is a blank graph and does not display any velocity data. - The lack of a plotted curve suggests that this graph either needs to be filled with corresponding velocity data or it could be an exercise for students to practice plotting velocity based on position data from the first graph. ### Interpretation and Relationships: Understanding the relationship between these two graphs is integral: 1. **Position to Velocity:** The velocity graph is the derivative of the position graph. For instance, when the position curve peaks (indicating maximum or minimum), the velocity curve would cross zero because the rate of change at these points is zero. 2. **Velocity to Position:** Conversely, the position graph can be derived by integrating the velocity graph. When velocity is zero, it indicates that the object is at an extreme (either maximum or minimum) or changing direction. This graphical representation helps in visualizing how an object’s motion changes over time and forms the basis of kinematic analysis in physics.
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