. Differential and integral calculus, an introductory course for colleges and engineering schools. d this means that the body rises for the first 5 seconds of its flight,comes to rest at A when t = 5, and then falls. From (b) we see that spasses O again 10 seconds after the start. Writing 5 for t in (b), we gets = —400, which is the height in feet to which the body rises. Writing10 for t in (a), we get v = 160, which is the velocity in feet per second withwhich the body passes O in its fall. This is the same velocity as thatwith which it rose from O, a result that could have been foreseen. Obs
. Differential and integral calculus, an introductory course for colleges and engineering schools. d this means that the body rises for the first 5 seconds of its flight,comes to rest at A when t = 5, and then falls. From (b) we see that spasses O again 10 seconds after the start. Writing 5 for t in (b), we gets = —400, which is the height in feet to which the body rises. Writing10 for t in (a), we get v = 160, which is the velocity in feet per second withwhich the body passes O in its fall. This is the same velocity as thatwith which it rose from O, a result that could have been foreseen. Observe that in each of these problems Acceleration = — = 32 feet per second. §147 APPLICATIONS OF INTEGRATION IN KINEMATICS 207 Problem 3. Harmonic Motion. A body is moving in a straight line and its velocity at time t is given bythe equation(a) v — — a sin fit, where a and n are known constants: determine the motion. dsSolution. We have -~ = — a sin nt. Integrating this,at /ds = | sin ^t did + C, or s = - cos nt + C. Let us choose the point from which s is measured so that s = - when. t = 0. Then - = - cos 0 + C, and C = 0. And the formula which ex- presses s in terms of t is (b) s = - cos /xJ. Formulae (a) and (b) completely describe the motion of the body. Let0 be the point from which s is measured, and let A be the bodys position when t = 0; then OA = -, and this is plainly the bodys greatest distance to the right of 0. Let A be the bodys position when t = - : then OA = - ^ and A; is the bodys greatest distance to theleft of 0. Let the student show from a study of equations (a) and (b) that the body, starting at A, moves to A and back again to A in—- units of time; that it repeats this oscillation in every succeeding — units of time; that at each passage through 0 it has its numerically greatest velocity of +aor —a; and that it passes through any point of its path in the same direction every —- units of time. The motion is termed harmonic. Such is the motio
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