Page:Elementary Text-book of Physics (Anthony, 1897).djvu/33

§ 20] $$t - t_{0}$$, multiplied by $$t - t_{0},$$, will represent the space traversed; hence or, since $$\frac{v}{2} = \frac{v_{0} + a (t - t_{0})}{2}$$, we have, in another form, {{MathForm2|(9a)|$$s - s_{0} = v_{0}(t - t_{0}) + \tfrac{1}{2} a (t - t_{0}).$$ Multiplying equations (4) and (9), we obtain {{MathForm2|(10)|$$v^2 = v^2_{0} + 2a (s - s_{0}).$$}} {{Img float | file    = Ant-Tbook-p19-fig9.png | width   = 250px | align   = right | cap     = Fig. 9 | capalign = center | alt     = }} When the point starts from rest, $$v_{0} = 0$$; and if we take the starting-point as the origin from which to reckon $$s$$, and the time of starting as the origin of time, then $$s_{0} = 0, t_{0} = 0$$, and equations (4), (9a), and (10) become $$v = at$$, $$s = \tfrac{1}{2}at$$, and $$v^2 = 2as$$.

Formula (9a) may also be obtained by a geometrical construction.

At the extremities of a line $$AB$$ (Fig. 9), equal in length to $$t - t_{0}$$, erect perpendiculars $$AC$$ and $$BD$$, proportional to the initial and final velocities of the moving point. For any interval of time $$Aa$$, so short that the velocity during it may be considered constant, the space described is represented by the rectangle $$Ca$$, and the space described in the whole time $$t - t_{0}$$, by a point moving with a velocity increasing by successive equal increments, is represented by a series of rectangles, $$eb$$, $$fc$$, $$gd$$, etc., described on equal bases, $$ab$$, $$bc$$, $$cd$$, etc. If $$ab, bc ...$$ be diminished indefinitely, the sum of the areas of the rectangles can be made to approach as nearly as we please the area of the quadrilateral $$ABCD$$. This area, therefore, represents the space traversed by the point, having the initial velocity $$v_{0}$$, and moving with the acceleration a during the time $$t - t_{0}$$. But $$ABCD$$ is equal to $$AC(t - t_{0}) + (BD - AC)(t - t_{0}) \div 2;$$ whence {{MathForm2|(9a)|$$s - s_{0} = v_{0}(t - t_{0}) + \tfrac{1}{2} a (t - t_{0}).$$}}

20. Angular Motion with Constant Angular Acceleration.— If a