Set 6 Problem number 7

• Problem

• Solution

Problem

Consider the region between the horizontal axis and the points ( 10 , 4 ) and ( 12 , 2 ) on a graph of the net force on an object, in Newtons, vs clock time in sec.  

• What is the area of this region? 
• What is the meaning of this area?

Solution

The area under the segment will consist of a trapezoid with altitudes 2 Newtons and 4 Newtons, and uniform width ( 12 sec - 10 sec) = 2 sec.

• The average of the altitudes represents the average of the forces 2 Newtons and 4 Newtons.

• If force is changing at a uniform rate the segment in fact represents the force vs. clock time precisely; otherwise it is only an approximation to the behavior of a curving graph.
• In general we therefore say that the average of the two forces is the approximate, not the exact, average force on the interval.

• The width of the trapezoid represents the time interval over which this approximate average force is sustained.
• The area of the trapezoid is the product of the average altitude, representing the approximate average force on the interval, and the width, representing the time interval.

• The Impulse-Momentum Theorem tells us that the product of net force and time interval is equal to change in momentum.
• The area therefore represents approximate average force * time interval = change in momentum on the object during the time interval.

The average altitude in the present example is ( 4 Newtons + 2 Newtons) / 2 = 3 Newtons.

• The area is thus 3 Newtons * 2 sec = 6 Newton seconds = 6 kg m/s.
• We interpret this as the approximate momentum change corresponding to the two points on the graph.