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Understanding Motion Through Graphical Analysis: Position, Velocity, and Acceleration

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This chapter delves into the graphical analysis of motion, focusing on three key scenarios: constant position (at rest), constant velocity, and constant acceleration. Each scenario has unique characteristics in terms of velocity and acceleration. At rest, both velocity and acceleration are zero, depicted as a horizontal line on the graph. For constant velocity, acceleration is zero, while the position graph takes a linear form. With constant acceleration, the acceleration remains constant, causing the velocity to change linearly, and the position graph forms a quadratic shape. Understanding these relationships through polynomial degrees, slopes, and areas enhances comprehension of motion.

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Understanding Motion Through Graphical Analysis: Position, Velocity, and Acceleration

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  1. Chapter 2 Legacy Graphical Analysis of Motion

  2. Describing Motion • First, it must be remembered that there are 3 different descriptions for motion • Constant position (at rest) • Constant velocity • Constant acceleration

  3. At Rest • For constant position or at rest, the velocity and acceleration are both 0 because the object is not moving. Position is a horizontal line because there is no change in position.

  4. Constant Velocity • For constant velocity the acceleration is zero because the object is not accelerating. The velocity, being constant, is represented by a horizontal line. The position is then a non-horizontal linear graph because the object is moving.

  5. Constant Acceleration • For constant acceleration the acceleration is a constant, obviously, so it is horizontal. The velocity then becomes linear because it is changing. The position-time graph then becomes quadratic, representing the non-constant speed of the object.

  6. Degrees • Something that may help is thinking of DT, VT, and AT in terms of polynomials • The degree of a polynomial is basically a number that refers to its shape • 0- constant (horizontal) • 1- linear • 2- quadratic • 3- cubic

  7. So for an object at rest all 3 graphs would have a degree of zero because they are horizontal • For constant velocity, velocity would be 0 and position would become 1 • For constant acceleration, acceleration would be 0, velocity would become 1, making position quadratic with a degree of 2 • For any of these quantities, each quantity below it increases by one degree from the degree of the initial quantity

  8. Area & Slope • One more thing to know is that the slope of each graph is equal to the value of the quantity above it (slope of VT = acceleration), while the area under a graph is equal to the value of the quantity preceding it (area under VT = position). It’s always helpful to keep this in mind.

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