§ 20]
MECHANICS OF MASSES.
19
, multiplied by
, will represent the space traversed; hence
|
(9)
|
or, since
, we have, in another form, {{MathForm2|(9a)|
Multiplying equations (4) and (9), we obtain
|
(10)
|
![](//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Ant-Tbook-p19-fig9.png/250px-Ant-Tbook-p19-fig9.png)
Fig. 9 When the point starts from rest,
; and if we take the starting-point as the origin from which to reckon
, and the time of starting as the origin of time, then
, and equations (4), (9a), and (10) become
,
, and
.
Formula (9a) may also be obtained by a geometrical construction.
At the extremities of a line
(Fig. 9), equal in length to
, erect perpendiculars
and
, proportional to the initial and final velocities of the moving point. For any interval of time
, so short that the velocity during it may be considered constant, the space described is represented by the rectangle
, and the space described in the whole time
, by a point moving with a velocity increasing by successive equal increments, is represented by a series of rectangles,
,
,
, etc., described on equal bases,
,
,
, etc. If
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
. This area, therefore, represents the space traversed by the point, having the initial velocity
, and moving with the acceleration a during the time
. But
is equal to
whence
|
(9a)
|
20. Angular Motion with Constant Angular Acceleration.— If a