. Differential and integral calculus, an introductory course for colleges and engineering schools. rter, and so these mean velocities approach moreand more nearly to being characteristic of the motion when t =3 seconds. The limit of this series of mean values is the charac-teristic we are seeking, and is termed the instantaneous velocitywhen t = 3. That limit may be found as follows: Let At be the time interval in seconds, following t = 3 the time (3 + At) seconds, the body falls 16(3 + A£)2 = (144 + 96 A* + 16 At2) feet, and the distance fallen in the interval At is (96 At + 16
. Differential and integral calculus, an introductory course for colleges and engineering schools. rter, and so these mean velocities approach moreand more nearly to being characteristic of the motion when t =3 seconds. The limit of this series of mean values is the charac-teristic we are seeking, and is termed the instantaneous velocitywhen t = 3. That limit may be found as follows: Let At be the time interval in seconds, following t = 3 the time (3 + At) seconds, the body falls 16(3 + A£)2 = (144 + 96 A* + 16 At2) feet, and the distance fallen in the interval At is (96 At + 16 A^2) the mean velocity during the interval At is 96 A* + 16 At2 A* = (96 + 16 At) feet per second. Finally, the limit of this as At = 0 is 96 feet per second. Thislimit is absolutely characteristic of the motion when t = 3 seconds,and is termed the instantaneous velocity at that time. Hence theinstantaneous velocity of a falling body 3 seconds after the fall beginsis 96 feet per second. Consider now the general case. Let a body start at O (either 106 DIFFERENTIAL CALCULUS §77. t+At line) and reach the point A in time t, and the point B in timet + At. Let OA = s and AB = As, so that the body moves the distance As in time At. Itsmean velocity during the time AsAt is -T- • When A2 and con- At sequently As are made verysmall, -r- is descriptive of the motion over a very short path AB, and so is very nearly charac- Asteristic of the motion at A. And the limit of -n as A£ = 0 is At absolutely characteristic of the motion at A. This limit, whichis Dts, we define to be the instantaneous velocity at A or at time t,or, more briefly, the velocity at A or at time t. We represent thisvelocity by v, so that we have Dts = v and -rr = Expressed in words, the definition runs: The instantaneous velocity at the instant t is the limit of the meanvelocity during a time interval At immediately following t, as At = is the derivative of the distance as to the time, or th
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