Set 6 Problem number 8

• Problem

• Solution

Problem

The graph below depicts the net force on an object vs clock time, with force in Newtons and clock time in seconds.  The gridlines depict units of 1.2 Newtons in the vertical direction and 8 seconds in the horizontal direction.

• What is the area under the graph for the time interval depicted, and what is the specific meaning of this area?

Solution

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

• The area of a trapezoid therefore represents a product of average force and elapsed time, which by the Impulse-Momentum Theorem represents change in the object's momentum.
• When summed the total area of the trapezoids therefore represents the approximate change in momentum of the objectduring the time interval.
• In the limit as the widths of the trapezoids approach zero the total area represents the exact change in momentum of the object.

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 Newtons * 8 sec = 9.6 Newton seconds = 9.6 kg m/s).

University Physics note that the precise momentum change or impulse would be obtained by integrating the function represented by the graph between the extreme times represented by the graph.