Set 6 Problem number 6

• Problem

• Solution

Problem

The graph below depicts velocity vs clock time, with velocity in meters/s and clock time in seconds.  The gridlines depict units of 1.2 m/s in the vertical direction and 5 seconds in the horizontal direction.

• What is the area under the graph between t = 2.4 sec and t = 7.2 sec, and what is the specific meaning of this area?
• What is the average slope under the graph between t = 3.6 sec and t = 9.6 sec, and what is the specific meaning of this slope?

Solution

The area under the curve could be broken into tiny trapezoids with altitudes representing velocities in m/s and widths representing time intervals.  The average of the altitudes of each trapezoid represents the approximate average velocity on that interval, and the width of the represents the time interval over which this approximate average velocity is sustained. 

• The area of a trapezoid therefore represents a product of average velocity and elapsed time, which represents change in position, also called the displacement.
• When summed the total area of the trapezoids therefore represents the approximate change in position from clock time t = 2.4 sec to t = 7.2 sec.
• In the limit as the widths of the trapezoids approach zero the total area represents the exact change in position.

We therefore estimate the area under the curve.

• We find the area by a trapezoidal approximation using several well-chosen trapezoids, or alternatively by counting the rectangles under the curve (the area of each rectangle representing the displacement 1.2 m/s * 5 sec = 6 m).

University Physics note that the precise position change or displacement would be obtained by integrating the function represented by the graph between t = 2.4 and t = 7.2 .|

The average slope between t = 3.6 sec and t = 9.6 sec is found by first calculating the rise between these points, which represents the change in velocity, and the run, which represents the time required for the velocity change.

• The meaning of the slope, which is the rise / run between the points, is change in velocity / change in clock time = average rate of velocity change = average acceleration between the two clock times.