Problem: On a graph of position vs time, with position in meters and time in seconds, we find the points ( 8, 9) and ( 5, 1). What do the rise and run between these points represent? What does the slope between these points represent?
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Solution: position will be plotted on the vertical axis with the independent variable time on the horizontal axis.
The rise will therefore indicate a change in position while the run will indicate a change in time. The rise represents a displacement in position from 9 to 1 meters or a displacement of
rise = -8 meters.
The run represents a change in time from 8 seconds to 5 seconds, which implies a time interval of
run = -3 seconds.
The slope is rise/run = -8 meters / ( -3 sec) = 2.666 meters/second. The units of this result are units of velocity, suggesting that the slope represents velocity. We see that in fact the slope was found by dividing a displacement by a time interval, which is congruent with the definition of velocity as change in position divided by elapsed time.
Generalized Response: On a graph of position s vs. clock time t, two points will have coordinates (t1, s1) and (t2, s2).
The rise between these points is from s1 to s2, a rise of `ds = s2 - s1. This rise represents the difference in position, or displacement, between position s1 and position s2.
The run is from t1 to t2, a run of `dt = t2 - t1. This run represents the difference in clock time between t1 and t2, or the time interval between t1 and t2.
The slope is the rise divided by the run, which is `ds / `dt, the position change divided by the time interval. This is the average rate at which position s changes with respect to time, or the average velocity of the object whose position is represented.
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Figure description: The graph below shows two points (t1, s1) and (t2, s2) on a graph of position vs. time. The rise is seen to be `ds = s2 - s1, representing the change in position. The run is seen to be `dt = t2 - t1, the time interval between the points.
The slope `ds / `dt therefore represents the position change divided by the time interval, which is the average rate at which the position changes. This average rate of change is generally called the average velocity.
