PhysicsLAB Resource Lesson
SVA: Slopes and Area Relationships

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Within these families there are patterns which relate one graph to another. The first pattern involves examining the slopes of the tangents drawn to a position-time graph which indicate the value of the object's instantaneous velocity. Dimensionally, if the units of the y-axis (m) are divided by the units of the x-axis (sec) the slope would have the dimension of velocity (m/sec).
 
position-time
s-t
slopes velocity-time
v-t
In this first example, the tangents have ever increasingly-positive slopes which indicate that the the object is gaining speed in a positive direction. When the numerical values for the slopes of these tangents are presented on a velocity-time graph, that graph would look like
In this second example, the tangents have ever decreasingly-negative slopes which indicate that the object is losing speed in a negative direction. When the numerical values for the slopes of these tangents are presented on a velocity-time graph, that graph would look like
 

A second pattern within these families of graphs involves the slope of the velocity-time graphs which indicate the value of the object's instantaneous acceleration. Dimensionally, if the units of the y-axis (m/sec) are divided by the units of the x-axis (sec) the slope would have the dimension of acceleration (m/sec2).
 
velocity-time
v-t
slopes acceleration-time
a-t
In this first example, the slope of the velocity-time graph is negative, indicating that the object is experiencing a negative acceleration while it is speeding up in a negative direction. Remember, since acceleration is a vector quantity it's value is influenced by both the change in the magnitude of the object's speed as well as the direction in which the object is traveling.
In this second example, the slope of the velocity-time graph is positive, indicating that the object is experiencing a positive acceleration while it is slowing down in a negative direction.
 

A third pattern within these families of graphs involves the area under the velocity-time graphs which indicates the object's  displacement during the time interval graphed. Dimensionally, if the units of the a-axis (sec) are multiplied by the units of the y-axis (m/sec) the area would have the dimension of displacement (m).
 
velocity-time
v-t
areas position-time
s-t
In this first example, the area bounded by the velocity function and the "time-axis" is composed of a triangle "sitting on top of"  a rectangle.  These two areas when added together represent the object's displacement during the time interval being graphed.  The area of the rectangle represents the object's displacement had it only been traveling at its final lower velocity; while, the area of the triangle represents the additional "distance traveled" because of its acceleration.
In this second example, the area bounded by the velocity function and the "time-axis" is composed of a rectangle "sitting on top of"  a triangle.  These two areas when added together represent the object's displacement during the time interval being graphed.  The area of the rectangle represents the object's displacement had it only been traveling at its original lower velocity; while, the area of the triangle represents the additional "distance traveled" because of its acceleration.
 

A fourth pattern within these families of graphs involves the area under an acceleration-time graph which indicates the change in the object's velocity during the time interval graphed. Dimensionally, if the units of the rectangle's base (sec) are multiplied by the units of its height (m/sec2) the area would have the dimension of velocity, or m/sec.

acceleration-time
a-t
areas velocity-time
v-t
In this first example, the area bounded by the acceleration function and the "time axis" is a rectangle.  Since the area is located in the IV quadrant, the change in the object's velocity is negative.  This event could occur either when the object gains speed in a negative direction or loses speed in a positive direction.
In this second example, the area bounded by the acceleration function and the "time axis" is a rectangle.  Since the area is located in the I quadrant, the change in the object's velocity is positive.  This event could occur either when the object gains speed in a positive direction or loses speed in a negative direction.



 
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