A partical traveIIing with uniform acceleration along a straight line has average velocities `v_(1),v_(2)` and `v_(3)` in successive time intervals `t_(1),t_(2)` and `t_(3)` respectively . Prove that, `(v_(2)-v_(1))/(v_(3)-v_(2))=(t_(1)+t_(2))/(t_(2)+t_(3))`

Explanation

Let the initial velocity of the particle be u and its acceleration be a. Also ,x,y and z are the velocities after the respective time intervals `t_(1),t_(2)"and"t_(3)`.
`:. x = u+at_(1),y = u+a(t_(1)+t_(2)), z = u +a(t_(1)+t_(2)+t_(3))`
Again `v_(1) = (u+x)/(2), v_(2) = (x+y)/(2), v_(3) = (y+z)/(2)`
`:. (v_(2)-v_(1))/(v_(3)-v_(2))=((1 )/(2)(x+y)-(1)/(2)(u+x))/((1)/(2)(y+z)-(1)/(2)(x+y))`
`" "=(y-u)/(z-x) = (u+a(t_(1)+t_(2))-u)/(u+a(t_(1)+t_(2)+t_(3))-(u+at_(1)))`
`" "=(t_(1)+t_(2))/(t_(2)+t_(3)).`