. Differential and integral calculus, an introductory course for colleges and engineering schools. tan • x — 1_ 2 v02 cos2 x\ The equation of OP is y = tan a • x. Solving these equations for x and y, we find, after easy reductions, the coordinates of P to be 2 y02 • / x 2v02 cos 0sm (0 — a),y= — — cos sin gcosa gcos a) tan a. §153 APPLICATIONS OF INTEGRATION IN KINEMATICS 219 Substituting these in the equation R = x cos a + y sin a, we have (8) Range on OP = R = 2 y„2 cos sin ( — a). gcosa To determine the value of which makes R a maximum, we setu = cos <£ sin ( — a);D^w = cos cos ( — «) —
. Differential and integral calculus, an introductory course for colleges and engineering schools. tan • x — 1_ 2 v02 cos2 x\ The equation of OP is y = tan a • x. Solving these equations for x and y, we find, after easy reductions, the coordinates of P to be 2 y02 • / x 2v02 cos 0sm (0 — a),y= — — cos sin gcosa gcos a) tan a. §153 APPLICATIONS OF INTEGRATION IN KINEMATICS 219 Substituting these in the equation R = x cos a + y sin a, we have (8) Range on OP = R = 2 y„2 cos sin ( — a). gcosa To determine the value of which makes R a maximum, we setu = cos <£ sin ( — a);D^w = cos cos ( — «) — sin 0 sin (<£ — «) = cos (2 — a). 2 7T ■ Q! 4+ 2 then When D+u = 0, 2 0 - « Then n = cos (| + |) sin(^ - |) = sin2 (| - |) = i[1-C0S(l°£)]=J(1Sina)- Therefore (9) Maximum range on OP = -^ :—— = Itf g COS a g(l + sin a) Setting 0 = - + a in this equation and p for the maximum range, we have (10) _2_ COS0. X Regarding p and 0 as variables, this is the polar equation of a parabolaturned downwards and having its focus at the pole 0. This parabolais the locus of points of maxi-mum range on all planesthrough 0 (a variable) attainedby projectiles fired from 0 withthe same velocity v0. It canbe proved that the parabolicpath of each such projectile istangent to the parabola (10).By revolving this parabola about its vertical axis, we get a surface whichis the locus of all points of maximum range reached by projectiles firedfrom 0 in all directions with the velocity v0. This surface is termed aparaboloid of revolution. A projectile fired from 0 with velocity v0 andin any direction would just reach this surface. All points inside thesurface can be hit by such a projectile and no point outside it. 220 INTEGRAL CALCULUS §154 154. Exercises. 1. The maximum range of a certain golf ball is 450 feet. What isthe velocity of the drive, how high does the ball rise, and how long is itin the air? 2. A cann
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