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vvЪЪЪЪйTopic #4(. Graphical Analysis of Motion
1. Position - Time Graph for Constant Velocity
2. Velocity from a Position - Time Graph
3. Position - Time Graph for a Complete Trip
4. Velocity - Time Graph for Constant Velocity
5. Velocity - Time Graph for Uniform Acceleration
6. Position - Time Graph for Uniform Acceleration
7. Acceleration - Time Graph for Uniform Acceleration
Notes should include:
Position - Time Graph for Constant Velocity: When an object moves with constant velocity, it covers the same distance each second, minute, hour, etc. The equation
d = vave t describes this motion. If direction is implied in the measurement of how far an object travels in a specified time interval the measurement is a statement of the object's displacement. Displacement is defined as the change in position an object experiences because it has moved. When it moves in a straight line d = vave t. Plot a graph of an object moving 60 m forward, as measured from a reference point each second of time. What does this graph look like? What is the shape of this graph?
Velocity from a Position - Time Graph: The calculation of slope allows you to calculate the rate of change between two variables. For example, if you calculate the slope of a graph like you produced above, you would obtain the velocity of the object. Velocity is the rate of change in position over time, much like speed is change in distance over time. In the above graph the slope is a constant because the function (the graph) is a straight line. What is the slope of the above graph? If your Algebra I skills are a bit rusty, remember that the slope of a straight line function (graph) is found by dividing the rise (change in the y value of the function) by the run (the change in the x value of the function). Be very careful not to drop the units in the calculation of slope, because they will combine in the ratio to form the new units for the slope value. Always remember that the slope represents a derived quantity, because it was derived from the x and the y values. In this case the velocity is being derived from the change in position divided by the change in time. The units for velocity in this situation are expressed as m/s. If the measurement is a velocity (a vector quantity) rather than a speed (a scalar quantity) the measurement would be followed by a statement of direction as in 60 m/s, forward.
Position - Time Graph for a Complete Trip: Displacement measurements require a reference point even when the motion is simply in a straight line. If an object moves forward its position value increases and the slope is positive. If it moves backwards its position value decreases and the slope is negative. Some circumstances may suggests an increase negatively, if it moved back past the reference point as it goes in the opposite direction from what is defined as forward. Because of this, displacement measurements will be either positive or negative. Also because of the use of a reference point the slopes (of position - time graphs can be positive, zero, or negative, depending on whether the object is moving forward, standing still, or is moving backwards.
Consider the following x,y coordinate pairs of data involving time and position.
0 s, 0 m ; 5 s, 100 m ; 10 s, 200 m ; 15 s, 200 m ; 20 s, 200 m ; 25 s, 150 m ; 45 s, 0 m.
On a sheet of graph paper, first make a two column data table of this data, placing the x data in the left column and the y data in the right column. Label the left column Time (s) and the right column Position (m). Second plot this data on a piece of graph paper. Label the x and the y axes with the name of the measurements and the units of measure used. Give the graph a title across the top. Finally, calculate the slope of this graph between 0 and 10 seconds, between 10 and 20 seconds, between 20 and 25 seconds, and between 25 and 45 seconds. Show each of these slope calculations on the same sheet of graph paper. Consider how your graph looks. Is it of reasonable size or did you try to mash it into a too small of a space for someone to clearly read it and for you to write in nice and neat labels. Anything that can't be read by a person evaluating your work would normally be considered wrong. It should be obvious to you as to why this would be so.
Velocity - Time Graph for Constant Velocity: Above you dealt with a situation in which the velocity changed from one time interval to another. Granted, it was a very abrupt change and not a smooth acceleration; none the less, it did change. Now suppose an object traveled at 50 m/s, E for 5 seconds. If you were to graph this as a velocity (y value) vs time (x value) what would the graph look like? Go ahead and graph it, using correct labels, title, and units. It should be obvious that the slope of this graph is zero. But what does this mean? First of all what units will the slope of a velocity - time graph have, regardless of whether the number is zero or something else? The slope of this graph is another derived quantity. What is the quantity you are deriving?
Also another important concept becomes useful when you are looking at a velocity - time graph. That concept involves calculating the area under the "curve" (the graph). It isn't all that unusual to use the word "curve" in place of a word like function or graph even when the graph is a straight line. Go ahead and calculate the area under this graph you have made. The shape of the area is a rectangle, so you are multiplying velocity by time (base times height. What measurement have you derived by doing this calculation? Do the units work out all right?
Velocity - Time Graph for Uniform Acceleration: Unlike the above example, where velocity was constant and there was no acceleration (its value was 0 m/s2), we will consider an example where the acceleration is uniform (that means constant) but greater in value than zero. Consider the motion of an object where it experiences a rate of uniform acceleration equal to 20 m/s2 (which is the same as 20 m/s/s). Plot the velocity - time graph for this situation for a time interval of 0 s to 5 s, using correct labels, units, and a title. What is the slope of this graph? Why is the line ascending instead of remaining horizontal like in the last graph? Calculate the area under the "curve" (graph) between 0 s and 2 s. What convenient geometric shape are you going to use to calculate this area? What is the name of the measurement that you are deriving by this calculation?
(Position - Time Graph for Uniform Acceleration: In the previous topic (topic #3), we had said that displacement can be found during uniform acceleration by the following equation d = vi t + 0.5 a t. If the uniform rate of acceleration is 60 m/s2, what would the displacement values be at one second intervals from 0 s to 5 s. Plot this displacement - time data using correct labels, units, and a title for the graph. If you were given this graph instead of producing it yourself, how might you calculate the slope at a specific point in order to determine the velocity at that point? Would you apply the concept of calculating slope of the line tangent to the point selected?
Acceleration - Time Graph for Uniform Acceleration: Plot an acceleration - time graph for a time interval from 0 s to 5 s, if the acceleration is uniform at a rate of 20 m/s2. If you calculate the area under this curve, what category of measurement would you be derived. What is the area under this curve?
Falling Bodies: Remember that as we saw in the last topic, topic 3, all problem solving and data processing skills associated with horizontal acceleration also applies to vertical motion. In this course we do not include the effect of air friction for moving objects in calculations, though we do discuss what its effect is on objects. The effect of air friction on moving objects that have both relatively large mass to volume ratios and low air drag coefficients is not going to interfere with learning the concepts associated with motion.
Vocabulary: Both Topic 2 and Topic 3 vocabulary.
Skills to be learned:
Plot Position - Time Graphs
Plot Velocity - Time Graphs
Plot Acceleration - Time Graphs
Calculate slopes of straight line graphs.
Calculate areas under "curves" (graphs)
Produce data tables (charts) from graphs
Assignments:
Textbook: Read / Study / Learn the examples in Chapter 5; Sections 5.1, 5.2, 5.3 and 5.4
WB Exercise(s): PS#3-2, PS#4-3, PS#4-4 (numbers 5 and 6), PS#4-5
Activities: TBA
Resources:
Topic Notes
Overhead and Board Notes discussed in class
Textbook: Chapter 5
Workbook: Lessons & Problem Sets:
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