. Differential and integral calculus, an introductory course for colleges and engineering schools. en(a) (1) a = Vai2 + a22, (2) <*i = a cos 0, a2 = a sin 0. 106. Resolution of Velocity along a Curve. Suppose a body which is in motion along a curve to be at the point P at time t ds(see first figure of Art. 103). Its tangential velocity at P is -j, and dx dvits velocities in the direction of the axes are -r: and -~. Now by dt dt J formuhe II and IV of Art. 101, <b) (1) it=v [it) +UJ (2) ^ -cos ^- ^=s,n • *• On comparing these equations with (a) above it appears that -r. dy ds and -j: are
. Differential and integral calculus, an introductory course for colleges and engineering schools. en(a) (1) a = Vai2 + a22, (2) <*i = a cos 0, a2 = a sin 0. 106. Resolution of Velocity along a Curve. Suppose a body which is in motion along a curve to be at the point P at time t ds(see first figure of Art. 103). Its tangential velocity at P is -j, and dx dvits velocities in the direction of the axes are -r: and -~. Now by dt dt J formuhe II and IV of Art. 101, <b) (1) it=v [it) +UJ (2) ^ -cos ^- ^=s,n • *• On comparing these equations with (a) above it appears that -r. dy ds and -j: are rectangular components of the tangential velocity -r • Now these three velocities are in magnitude proportional to andin direction identical with dx, dy, and ds, the sides of the triangle dxPBT. We may so choose PB that PB = -j, and then the three sides of PBT represent in magnitude and direction the tangential ds dx dii velocity -r and its components along the axes -r- and ~ • 106 SIMPLE FORMULA OF KINEMATICS 151 Again, dividing by dt formulae V of Art. 102, we have dsdt do ds p*=8m**. These equations show that p -r, and -r are rectangular velocity components of the tangential ve- dslocity -r. Now the direction of ds ■rr is along the tangent to the path at P, and -=- is plainly along p. Consequently p -r. is along a per- 0 pendicular to p at P, that is, alonga tangent at P to the circular arc described about 0 with p asradius. This follows also from Art. 79. These three velocitiesare shown in the figure in direction but not in magnitude. Example. A point on the circumference of a circle of radius r traversesthe curve with a constant (tangential) velocity of k feet per second. Find the velocities of the projectionsof the point on two perpendicularlines in the plane of the circle. Solution. We choose for axes linesthrough the center of the circle par-allel to the given lines. Then thevelocities of the point upon the givenlines are the same as upon the axes of coordinat
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1912
