Problem: On a graph of kinetic energy vs speed, with kinetic energy in Joules and speed in mph, we find the points ( 5, 1) and ( 8, 15). 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 kinetic energy vs. speed, the dependent variable kinetic energy will be plotted on the vertical axis with the independent variable speed on the horizontal axis. The rise will therefore indicate a change in kinetic energy while the run will indicate a change in speed. The rise represents a change in kinetic energy from 1 to 15 Joules, or an kinetic energy change of 14 Joules. The run represents a change in speed from 5 mph to 8 mph, which implies a speed change of 3 mph. The slope is rise/run = 14 Joules / ( 3 mph) = 4.666 Joules / mph = 4.666 J / mph. Note that this information might tell us how much energy must be added per mph to increase the speed of an object.
Generalized Response: On a graph of kinetic energy vs. speed v, two points will have coordinates (v1, KE1) and (v2, KE2).
The rise between these points is from KE1 to KE2, a rise of `dKE = KE2 - KE1. This rise represents the difference in kinetic energy between initial kinetic energy KE1 and final kinetic energy KE2.
The run is from v1 to v2, a run of `dv = v2 - v1. This run represents the difference in speed between v1 and v2.
The slope is the rise divided by the run, which is `dKE / `dv, the kinetic energy change divided by the change in speed. This is the average rate at which kinetic energy KE changes with respect to speed.
We can imagine for example that the kinetic energy of an automobile is determined at a variety of speeds, perhaps by accelerating the automobile with known net forces through known distances. The slope at a give velocity of the resulting model will tell us the approximate kinetic energy increase required for a given small velocity increase.
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Figure description: The graph below shows two points (t1, KE1) and (t2, KE2) on a graph of kinetic energy vs. time. The rise is seen to be `dKE = KE2 - KE1, representing the change in kinetic energy. The run is seen to be `dv = v2 - v1, the time interval between the points.
The slope `dKE / `dv therefore represents the kinetic energy change divided by the time interval, which is the average rate at which the kinetic energy changes with respect to velocity.
This average rate of change might for example tell a curious driver something about how much energy is required to increase his or her speed by a given amount (though the quantity obtained here would indicate only the energy required for the speed increase alone, and would not take account of such things as friction and air resistance).
