A point move with uniform acceleration and `upsilon_(1), upsilon_(2)` and `upsilon_(3)` denote the average velocities in the three successive intervals of time `t_(1), t_(2)` and `t_(3)`. which of the following relations is correct ?

Options

A. `(v_(1)-v_(2)):(v_(2)-v_(3)=(t_(1)-t_(2):(t_(2)+t_(3)`.
B. `(v_(1)-v_(2)):(v_(2)-v_(3)=(t_(2)-t_(2):(t_(2)+t_(3)`
C. `(v_(1)-v_(2)):(v_(2)-v_(3)=(t_(1)-t_(2):(t_(2)+t_(3)`
D. `(v_(1)-v_(2)):(v_(2)-v_(3)=(t_(1)-t_(2):(t_(2)+t_(3)`

Answer

B

Explanation

Suppose `u` be the initial velocity.
Velocity after time `t_(1)`: `v_(11) =u+at_(1)`
Velocity after time `t_(1) +t_(2)`: `v_(22) =u +a (t_(1) +t_(2)`
Velocity after time `t_(1) +t_(2) +t_(3)`:
`v 3 =u+a (t_(1) +t_(2) +t_(3))`
Now `v_(1) =(u+v_(11))/(2) =(u+u+at_(1))/(2) =u+(1)/(2) at_(1)`
`v_(2) =(v_(11) +v_(22))/(2) =u+at_(1) +(1)/(2_(2))`
`v_(3) =(v_(33)+ v_(33))/(2) =u+at_(2) + (1)/(2) at_(3)`
So `v_(1)-v_(2) =- (1)/(2) a(t_(1) +t_(2))`
`v_(2)-v_(3) =-(1)/(2) a(t_(2)-t_(3))`
`(v_(1)-v_(2))`: `(v_(2)-v_(3) =(t_(1) +t_(2))` : `(t_(2) +t_(3))`.