. Applied calculus; principles and applications . d with a given velocity vq. Using again gf/c^, as thecoefficient of resistance, the resistance of the air on a particlefor a unit of velocity, and taking the particle of unit mass,with g constant, g=^g; (1) whence, ttt^ = —kgdt; integrating, tan-MA;-,-j = tan-^ (kvo) — kgt, 460 INTEGRAL CALCULUS where C = tan^ (kvo); solving, V = ds _ 1^ kvp — tan kgt dt~ k 1 -{-kvotsinkgt ^^^ which gives the velocity in terms of the time. To get it interms of the space; from (1), ±M1 -2gk^ds; integrating, log ^+K-:J 1 + kWwhere c = log (1 + A;W); whence,
. Applied calculus; principles and applications . d with a given velocity vq. Using again gf/c^, as thecoefficient of resistance, the resistance of the air on a particlefor a unit of velocity, and taking the particle of unit mass,with g constant, g=^g; (1) whence, ttt^ = —kgdt; integrating, tan-MA;-,-j = tan-^ (kvo) — kgt, 460 INTEGRAL CALCULUS where C = tan^ (kvo); solving, V = ds _ 1^ kvp — tan kgt dt~ k 1 -{-kvotsinkgt ^^^ which gives the velocity in terms of the time. To get it interms of the space; from (1), ±M1 -2gk^ds; integrating, log ^+K-:J 1 + kWwhere c = log (1 + A;W); whence, = -2 -=(fj= 1 v,^e-^9k^s _ ^^ (1 g-2 gk^s\ ^ Writing tan kgt in (2) in terms of sine and cosine andintegrating, s = T^ log {kvo sin kgt + cos kgt),K g which gives the space described by the particle in terms of the time. 235. Curvilinear Motion. — Let a body slide without friction down any curve ah. Theacceleration caused by gravityat any point P is ^ sin a, wherea = PTD, PT being a tangentto the curve. Let PT = ds; then -PD =dy; ds = gsma — n^y (1) d^2 ^— ^ds Let t/o be the ordinate of the initial point on the curve; thenV = 0 when y = yo. SIMPLE CIRCULAR PENDULUM 461 Integrating (1) gives ^=Jl = ^2g(2/o-2/). (2) It follows from (2) that the velocity of a body acquiredby moving freely down any frictionless path is the same,and is what it would acquire in falling freely through thevertical height between the initial and terminal points. /ds—, the time will depend upon the path. 236. Simple Circular Pendulum. — Consider the motionof a particle on a smooth circular arc under the actionof gravity as the only force.
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