 Graphing & Vector Quantities:

Objectives:

1.    Explain how graphs can be used to describe relationships between time, distance, and speed.

2.    Given a graph, the student will be able to determine whether acceleration, constant speed, or deceleration is occurring.

3.    Distinguish between a scalar and a vector quantity.  (Be able to identify examples.)

4.    Draw vector diagrams for velocities and be able to solve simple situations involving vector addition and subtraction.

Key Terms:

scalar quantity    vector quantity    resultant    vector    resolution

Graphs are traditionally used to visually express the relationship between two or more different but related quantities.  The graph is constructed using the x-axis (horizontal) as the independent predictable reference and the y axis (vertical) as the dependant measured variable.    You will notice in the three graphs that follow, while all displaying the same situation, the independent value (time) remains constant while the dependant value changes the shape and the meaning of the graph.  The examples used is compare speed, velocity, & acceleration.  The SI units of speed and velocity are m/s while acceleration is m/s2.

 Speed (m/s) time (s) 10 1 20 2 30 3 40 4 50 5

When labeling your charts and graphs it important to use the proper units.  The connecting line on your graph points should be the "best fit" or line bisecting the median between the data points. This is a graph depicting constant acceleration and is displaying speed over time.  As you can see, the graph is characterized by a straight line with a positive slope (slant increasing from left to right).  The slope gives you a good idea of the rate of acceleration.  The graph displays the property of linearity meaning that the variables used are in direct proportion.

 Distance (m) time (s) 5 1 20 2 45 3 80 4 125 5 180 6 245 7 The graph is now dealing with the distance covered by an object during free fall.  You will notice that the line now is parabolic (curves upward) with an increasing positive slope.

 acceleration (m/s2) time (s) 9.8 1 9.8 2 9.8 3 9.8 4 9.8 5 9.8 6 The graph now displays the acceleration during free fall.  The slope is now zero neither increasing nor decreasing.  Acceleration is constant.

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A vector quantity is one that requires both magnitude and direction to adequately describe it.  A scalar quantity, though, requires only magnitude.  Examples of vectors include velocity, acceleration, weight, and momentum.  Examples of a scalar unit would include speed, mass, volume, and time neither of which requires a direction to adequately it.

Vectors can be displayed using vector diagrams.  Consider the following example.

50m/s   5m/s wind

What would be the result of the two opposing vectors? [

If the wind switched to a tail wind the result would be 55m/s. [

Vectors can also be directed at angles.  Consider this plane and the wind traveling at right angles to each other. The black lines are the vertical and horizontal components of the final velocity (green).   The blue line is the resultant or the diagonal of the rectangle described by the two vectors.  The resultant can be calculated using the following formula:

(resultant)2 = (vertical)2 + (horizontal)2

(5)2 = (4)2 + (3)2

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