Three Chapter 2

• Given the position-time (d-t) graph above, one can construct the velocity-time (v-t) graph above. To do so, follow the following steps:
  • Ask what kind of line do I have on the given graph? ÞStraight horizontal
  • Ask what kind of motion does this line represent? Þ The object is stopped.
  • Ask what kind of line on the graph to be produced represents the type of motion? Þ Straight horizontal on the time axis
  • Sketch in the line.
• Following the same series of steps outlined above, section B is a straight diagonal line. This line represents uniform positive motion because the slope is positive. On the v-t graph, this motion is represented by a straight horizontal line above the time axis, when extended to the vertical axis, intercepts at the slope of the d-t graph. The line is already sketched.
• You should try to follow this procedure for sections C, D, and E.
• This procedure can be used for converting not only a d-t graph to a v-t graph, but from ANY given graph to ANY desired graph. Learn this general principle!
• The example above had all straight lines. If there are curved lines, then we must take the slope of the tangent to the curve at various times to make a table of velocity versus time. The data in this table can be used to plot a v-t graph.
• Acceleration - The rate of change of velocity.
  • Uniform acceleration is a special case of acceleration.
• Uniform Acceleration - Equal changes in velocity in equal time intervals.
  • produces a straight diagonal line on a v-t graph, and a parabolic line on a d-t graph.
• The slope of the v-t graph is acceleration.
• Note these possible situations for graphs.

Not Moving Moving + Moving - accelerating + accelerating -

• Uniform Motion
  • The position time graph is a straight diagonal line.
  • Because the velocity never changes, the velocity time graph is a straight horizontal line, intercepting the vertical axis at the speed.

• Consider an object moving at +3m/s uniformly. Therefore distance traveled from 0 to 5 s is:

• The velocity-time graph for this motion is:

• From 0 to 5 s, the area between the line and the time axis is given by:

• \ (therefore) the area between the line and the time axis on a velocity-time graph, between two times is the displacement change during that time interval.
• This principle applies to uniform or non uniform motion.
• Given the following v-t graph, what is the displacement between 5 and 20 s?

• To answer the question, one has to find the area between 5 and 20 s, and between the line on the graph and the time axis. This area can be broken up into two areas, the area of a triangle and the area of a rectangle.

• The area of the triangle is A = (1/2)bh = (1/2)(20s - 5s)(40m/s - 20m/s) = 150 m
• The area of the rectangle is A = LW = (20s - 5s)(20m/s - 0m/s) = 300 m
• The total area is therefore the sum of these two A_T = 150 m + 300 m = 450 m
• \ (therefore) the displacement between 5 and 20 s is 450 m.
• Area above the time axis is positive and below the time axis is negative. The sign is important to finding the total displacement.
• Units must be included (found from units on axis).
• For the given v-t graph, sketch the corresponding d-t graph.

• By applying the previously discussed method:
  • Ask what kind of line do I have on the given graph?
  • Ask what kind of motion does this line represent?
  • Ask what kind of line on the graph to be produced represents the type of motion?
  • Sketch in the line.

and the concept that area between the line and the time axis between two times is the

change in position, the following position-time graph can be sketched.

• If you cannot produce this graph by applying the above method, then make a table of values of area between the line and the time axis versus time. Now plot the graph.
• For curved v-t graphs, area is still displacement. To find the area, two methods can be used.
  • divide the area up into squares and rectangles to approximate the total area.
  • Use Integral Calculus to find the area. Þ Of course, if you don't know how to do this, use the first method!

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October 28, 2013