Higher Concepts of MathematicsCALCULUSIntegralsand Summationsin PhysicalSystemsDifferentials and derivatives arose in physical systems when small changes in one quantity wereconsidered. For example, the relationship between position and time for a moving object led tothe definition of the instantaneous velocity, as the derivative of the distance traveled with respectto time, ds/dt. In many physical systems, rates of change are measured directly. Solvingproblems, when this is the case, involves another aspect of the mathematics of dynamic systems;namely integral and summations.Figure 7 is a graph of the instantaneous velocity of an object as a function of elapsed time. Thisis the type of graph which could be generated if the reading of the speedometer of a car wererecorded as a function of time.At any given instant of time, the velocityFigure 7 Graph of Velocity vs. Timeof the object can be determined byreferring to Figure 7. However, if thedistance traveled in a certain interval oftime is to be determined, some newtechniques must be used. Consider thevelocity versus time curve of Figure 7.Let's consider the velocity changesbetween times tA and tB. The firstapproach is to divide the time interval intothree short intervals , and to(D t1,D t2,D t3)assume that the velocity is constant duringeach of these intervals. During timeinterval Dt1, the velocity is assumedconstant at an average velocity v1; duringthe interval Dt2, the velocity is assumedconstant at an average velocityv2; duringtime interval Dt3, the velocity is assumedconstant at an average velocityv3. Thenthe total distance traveled is approximately the sum of the products of the velocity and theelapsed time over each of the three intervals. Equation 5-10 approximates the distance traveledduring the time interval from tato tb and represents the approximate area under the velocity curveduring this same time interval.s= v1Dt1+ v2Dt2+ v3Dt3(5-10)Rev. 0 Page 41MA-05
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