. Differential and integral calculus. nd solving for dt, we have \2gr2) sds Vv Hence, /=(^)*j(,0,-^-| 48 1374 vers1— I + C. Sn \ 374 Integral Calculus Let s = s0 when / = o; then C=(^fS-0Yers-i2=(j^f^.\2gr2J 2 \2gr2) 2 Hence, -(ife/l^-^-i vers * 1 S0 2 (?) which gives the time / for a body to fall from height s0 toheight s. Cor. i. If in (i) we make s0 = oo and s= r, we have v = \l2gr for the velocity with which a body would strike the earth if itfell from an infinite distance in a vacuum. Since g = 32 ft., nearly, and r= 20,900,000 ft., nearly, wefind V = 7 miles per second, nearly. Cor. 2. I
. Differential and integral calculus. nd solving for dt, we have \2gr2) sds Vv Hence, /=(^)*j(,0,-^-| 48 1374 vers1— I + C. Sn \ 374 Integral Calculus Let s = s0 when / = o; then C=(^fS-0Yers-i2=(j^f^.\2gr2J 2 \2gr2) 2 Hence, -(ife/l^-^-i vers * 1 S0 2 (?) which gives the time / for a body to fall from height s0 toheight s. Cor. i. If in (i) we make s0 = oo and s= r, we have v = \l2gr for the velocity with which a body would strike the earth if itfell from an infinite distance in a vacuum. Since g = 32 ft., nearly, and r= 20,900,000 ft., nearly, wefind V = 7 miles per second, nearly. Cor. 2. If in (2) we make s == r, we have the time for abody falling from the height ^0 to the earth. CURVILINEAR MOTION. 238. Velocity of a body down a curve in a vertical plane. Let ST be any curve in the plane VOX, referred to OY and OX 2,s axes, OF being posi-tive downwards. Let P bethe position of the body atany instant and let PA =(g) represent the accelerationdue to gravity. Draw AC_L to the tangent PB, andFig. 61. let PAC = 6 ; then. PC = g sin $ = acceleration in direction of motion. Mechanical Applications 375 But if we let PB = ds, then PA = dy; hence -^ = sin 0. ds Hence, § 82 d2s a¥ ~~ dsd2s df- Jy- = gds = gdy. Integrating between limits y andy we have ds2 2dt2^ g(y- ?/) (0 .-. § 17, v*= 2g(y-/). Comparing the last equation with (c) § 233, Cor., we seethat the velocity acquired by a body in rolling down a curve isthe same as it would acquire in falling freely through the verticalheight. Cor. From (1) we have ds ds dy dt ^2g(y- /) dy ^2g(y- y^f> ds dy .. /= I -r , • (2) J ) W We are to find what this expression becomes when appliedto the cycloid. 376 Integral Calculus From the equation of the cycloid, x — a vers-1- — V2 ay — y*, we obtain dx y
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