A point moves with uniform acceleration and v1, v2, and v3 denote the average velocities in the three successive intervals of time t1, t2, and t3 Which of the following relations is correct?

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A point moves with uniform acceleration and v₁, v₂, and v₃ denote the average velocities in the three successive intervals of time t₁, t₂, and t₃ Which of the following relations is correct?

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$(v_{1} - v_{2}) : (v_{2} - v_{3}) = (t_{1} - t_{2}) : (t_{2} + t_{3})$

$(v_{1} - v_{2}) : (v_{2} - v_{3}) = (t_{1} - t_{2}) : (t_{2} - t_{3})$

$(v_{1} - v_{2}) : (v_{2} - v_{3}) = (t_{1} + t_{2}) : (t_{2} + t_{3})$

$(v_{1} - v_{2}) : (v_{2} - v_{3}) = (t_{1} - t_{2}) : (t_{1} - t_{3})$

answer is B.

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Detailed Solution

Suppose u be the initial velocity.

$Velocity after time t_{1} : v_{11} = u + a t_{1}$

$Velocity after time t_{1} + t_{2} : v_{22} = u + a (t_{1} + t_{2})$

$Velocity after time t_{1} + t_{2} + t_{3} :$

$\begin{array}{l} v_{33} = u + a (t_{1} + t_{2} + t_{3}) \\ Now v_{1} = \frac{u + v_{11}}{2} = \frac{u + u + a t_{1}}{2} = u + \frac{1}{2} a t_{1} \\ v_{2} = \frac{v_{11} + v_{22}}{2} = u + a t_{1} + \frac{1}{2} a t_{2} \\ v_{3} = \frac{v_{22} + v_{33}}{2} = u + a t_{1} + a t_{2} + \frac{1}{2} a t_{3} \end{array}$

$\begin{array}{l} So v_{1} - v_{2} = - \frac{1}{2} a (t_{1} + t_{2}) \\ v_{2} - v_{3} = - \frac{1}{2} a (t_{2} + t_{3}) \\ (v_{1} - v_{2}) : (v_{2} - v_{3}) = (t_{1} + t_{2}) : (t_{2} + t_{3}) \end{array}$