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S E C T I O N 2 . 7 • Kinematic Equations Derived from Calculus 45 duration $t n . From the definition of average velocity we see that the displacement during any small interval, such as the one shaded in Figure 2.15, is given by where is the average velocity in that interval. Therefore, the displacement during this small interval is simply the area of the shaded rectangle. f # t i is the sum of the areas of all the rectangles: where the symbol + (upper case Greek sigma) signifies a sum over all terms, that is, i to t f . Now, as the intervals are made smaller and smaller, the number of terms in the sum in- n : 0, the displacement is (2.14) or Note that we have replaced the average velocity with the instantaneous velocity v xn in the sum. As you can see from Figure 2.15, this approximation is valid in the limit of x -t graph for motion along a straight line, we can obtain the displacement during any time interval by measuring the area The limit of the sum shown in Equation 2.14 is called a definite integral and is written (2.15) where v x (t) denotes the velocity at any time t. If the explicit functional form of v x (t) is known and the limits are given, then the integral can be evaluated. Sometimes the v x -t graph for a moving particle has a shape much simpler than that shown in Figure 2.15. xi . In this case, the v x -t graph is a horizontal line, as in Figure 2.16, and the displacement of the particle dur- As another example, consider a particle moving with a velocity that is proportional to t, x ! a x t, where a x is the constant of proportionality (the $ x ! v xi
$ t
(when v x ! v xi ! constant) lim $ t n : 0 % n v xn
$ t n ! & t f t i v x (t)dt v xn Displacement ! area under the v x -t graph $ x ! lim $ t n : 0 % n v xn $ t n $ x ! % n v xn $ t n v xn $ x n ! v xn $t n v x = v xi = constant t
f v xi t ∆t t
i v x v xi Figure 2.16 The velocity–time curve for a particle moving with constant velocity v xi . The displacement of the particle during the time interval t f # t i is equal to the area of the shaded rectangle. Definite integral |