A point moves with a uniform acceleration and ` v_1 v_2 v_3` denote the average velociies in the three succellive intervals of time ` _1 , t_2` and t_3`. Find the ration of ( v-1 - v_2) and ( v_2 - v_3).

Explanation

Let the point starts moving from (O) with a uniform acceleration (a) along a st. line . It reaches at locations (A). (B) and (C ) at timeings ` t_1, t-_2, and t_3` respectively Fig. 2 ( HT). 6.
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Let ` v'. V'', v''` be the velocity of point at ` A, B` and ` C` respectively
:. Initial velocity , ` v=u = a t`
Using the formaula , v=u + at`
Taking motion from ` O` to ` A` we have , ` v' = 0 = a t_1 =a t-1`
Taking motion from ` O` to ` B` we have , v'' = 0 + a (t-1 + t-2) = a (t_1 +t_2)`
Taking motion from ` O` to C` we have , v''' = 0 + a (t-1 + t_3 ) = a (t_1 + t_2 + t_3)`
:. Avetage velocity in interval of time ` t_1`.
` v_1 (0+v')/2 ( at_10/2`
Average velocity in interval of time ` t_2,
`v_2 =(v'+v'')/2 = (at _1 + a (t_1+ t_2)/2 = at _2`
Average velocity in interval of time ` t_3`,
`v_3 = (v'' + v'')/2 = (a(t_1 +t_2) + a (t_1 + t_2 + t_3)/2 =a (t_1 =t_2) + 1/2a t_3`
:. ` v_2 -v_1 = ( at-1 + 1/2 at_2) - (at-1)/2 a/2 (t_1 + t_2)` ...(4)
` and ` v_3 -v_2 = [a (t_1 + t_2) + 1/2 at_3] - [at_1 + 1/2 at_2] = a/2 (t_2 + t_3)` ...(5
From (4) and (5) , ` (v_2 -v_1)/(v_3-v_2) = ((t_1 + t_2))/((t_2+t_3))` or ` (v_1 0v_2)/(v_2-v_3) = (t_1+ t_2)/(t_2+t_3)` .