Set 6 Problem number 7

• Problem

• Solution

Problem

On a graph of the net force on an object vs clock time, with force in Newtons and clock time in seconds, what is the area between the horizontal axis and the points ( 4 , 13 ) and ( 9 , 18 )?  What is the meaning of this area?

Solution

The area under the segment will consist of a trapezoid with altitudes 18 Newtons and 13 Newtons, and uniform width ( 9 sec - 4 sec) = 5 sec.

• The average of the altitudes represents the average of the forces 18 Newtons and 13 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 ( 13 Newtons + 18 Newtons) / 2 = 15.5 Newtons.

• The area is thus 15.5 Newtons * 5 sec = 77.5 Newton seconds = 77.5 kg m/s.
• We interpret this as the approximate momentum change corresponding to the two points on the graph.