Problem: On a graph of velocity vs time, with velocity in meters/second and time in seconds, we find the points ( -2, -4) and ( 7, -9). What do the rise and run between these points represent? What does the slope between these points represent?
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Solution: Since the graph is of velocity vs. time, the dependent variable velocity will be plotted on the vertical axis with the independent variable time on the horizontal axis.
The rise will therefore indicate a change in velocity while the run will indicate a change in time. The rise represents a change in velocity from -4 to -9 meters/second or a velocity change of
rise = -5 meters/second.
The run represents a change in time from -2 seconds to 7 seconds, which implies a time interval of 9 seconds. The slope is rise/run = -5 meters/second / ( 9 sec) = -.5555 meters/second/second.
The units of this result are units of acceleration, suggesting that the slope represents acceleration. We see that the slope was found by dividing a velocity change by a time interval, which is congruent with the definition of acceleration as change in velocity divided by elapsed time.
Generalized Response: On a graph of velocity v vs. clock time t, two points will have coordinates (t1, v1) and (t2, v2).
The rise between these points is from v1 to v2, a rise of `dv = v2 - v1. This rise represents the difference in velocity, or displacement, between velocity v1 and velocity v2.
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 `dv / `dt, the velocity change divided by the time interval. This is the average rate at which velocity v changes with respect to time, or the average acceleration of the object whose velocity is represented.
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Figure description: The graph below shows two points (t1, v1) and (t2, v2) on a graph of velocity vs. time. The rise is seen to be `dv = v2 - v1, representing the change in velocity. The run is seen to be `dt = t2 - t1, the time interval between the points.
The slope `dv / `dt therefore represents the velocity change divided by the time interval, which is the average rate at which the velocity changes. This average rate of change is generally called the average acceleration.
