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

$t-t_{0}$, multiplied by $t-t_{0},$, will represent the space traversed; hence

$s-s_{0}={\frac {v+v_{0}}{2}}(t-t_{0});$

(9)

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

$v^{2}=v_{0}^{2}+2a(s-s_{0}).$

(10)

Fig. 9 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

$s-s_{0}=v_{0}(t-t_{0})+{\tfrac {1}{2}}a(t-t_{0}).$

(9a)

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