A point moves...

Question

A point moves with uniform acceleration and $v_{1}, v_{2}$ and $v_{3}$ 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?

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Solution

The correct option is B

Let $u_{1}, u_{2}, u_{3}$ and $u_{4}$ be velocities at time $t = 0, t_{1}, (t_{1} + t_{2})$ and $(t_{1} + t_{2} + t_{3})$ respectively and let the acceleration be $a$ then

$v_{1} = \frac{u_{1} + u_{2}}{2}, v_{2} = \frac{u_{2} + u_{3}}{2}$ and $v_{3} = \frac{u_{3} + u_{4}}{2}$

Also $u_{2} = u_{1} + a t_{1}, u_{3} = u_{1} + a (t_{1} + t_{2})$

and $u_{4} = u_{1} + a (t_{1} + t_{2} + t_{3})$

By solving, we get $\frac{v_{1} - v_{2}}{v_{2} - v_{3}} = \frac{t_{1} + t_{2}}{t_{2} + t_{3}}$

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