Graphs of Motion

Practice

practice problem 1

Complete the worksheet on the first page of worksheet-compare.pdf. Fill each grid space with an appropriately concise answer.

solution

The answers are on the second page of worksheet-compare.pdf.

page 1: the questions page 2: the answers

practice problem 2

Work along with this example using worksheet-transform.pdf. The graph below shows velocity as a function of time for some unknown object.

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  1. What can we say about the motion of this object?
  2. Plot the corresponding graphs of displacement and acceleration as functions of time.

solution

  1. The problem presents us with a velocity-time graph. Do not read it as if it was showing you position. You can't immediately determine where the object is from this graph. You can say what direction it's moving, how fast it's going, and whether or not it's accelerating, however. The motion of this object is described for several segments in the animation below.

    [slide]

  2. Acceleration is the rate of change of displacement with time. To find acceleration, calculate the slope in each interval.

    [slide]

    Plot these values as a function of time. Since the acceleration is constant within each interval, the new graph should be made entirely of linked horizontal segments.

    [slide]

  3. Displacement is the product of velocity and time. To find displacement, calculate the area under each interval.

    [slide]

    Find the cumulative areas starting from the origin (given an initial displacement of zero)

     0 s →  0                                =   0 m
     4 s →  0 + 8                            = + 8 m
     8 s →  0 + 8 - 8                        =   0 m
    12 s →  0 + 8 - 8 - 16                   = -16 m
    16 s →  0 + 8 - 8 - 16 - 8               = -24 m
    20 s →  0 + 8 - 8 - 16 - 8 + 0           = -24 m
    24 s →  0 + 8 - 8 - 16 - 8 + 0 + 8       = -16 m
    30 s →  0 + 8 - 8 - 16 - 8 + 0 + 8 + 24  = + 8 m

    Plot these values as a function of time. Pay attention to the shape of each segment. When the object is accelerating, the line should be curved.

    [slide]

practice problem 3

Sketch the displacement-time, velocity-time, and acceleration-time graphs for …
  1. an object moving with constant velocity. (Let the initial displacement be zero.)
  2. an object moving with constant acceleration. (Let the initial displacement and velocity be zero.)

solution

  1. Since the velocity is constant, the displacement time graph will always be straight, the velocity-time graph will always be horizontal, and the acceleration-time graph will always lie along the horizontal axis. When the velocity is positive, the displacement should have a positive slope. When the velocity is negative, the displacement should have a negative slope. When the velocity is zero, all the curves should be horizontal.

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  2. Since the acceleration is constant, the displacement time graph will always be a parabola, the velocity-time graph will always be straight, and the acceleration-time graph will always be horizontal. When the acceleration is positive, the velocity should have a positive slope, and the displacement should bend upward. When the acceleration is negative, the velocity should have a negative slope, and the displacement should bend downward. When the acceleration is zero, all the curves should be horizontal.

    [slide]