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

• A. $(v_{1} - v_{2}):(v_{2} - v_{3}) = (t_{1} - t_{2}):(t_{2} + t_{3})$
• B. $(v_{1} - v_{2}):(v_{2} - v_{3}) = (t_{1} + t_{2}):(t_{2} + t_{3})$
• C. $(v_{1} - v_{2}):(v_{2} - v_{3}) = (t_{1} - t_{2}):(t_{1} - t_{3})$
• D. $(v_{1} - v_{2}):(v_{2} - v_{3}) = (t_{1} - t_{2}):(t_{2} - t_{3})$

Answer 1

Answer: (B)

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

Answer 2

Let $u_{1},u_{2},u_{3}$ and $u_{4}$ be velocities at time

$t = 0,\mspace{6mu} t_{1},\mspace{6mu}(t_{1} + t_{2})$ and $(t_{1} + t_{2} + t_{3})$ respectively and

acceleration is a then

$v_{1} = \frac{u_{1} + u_{2}}{2},\mspace{6mu} v_{2} = \frac{u_{2} + u_{3}}{2}\text{and}\mspace{6mu} v_{3} = \frac{u_{3} + u_{4}}{2}$

Also $u_{2} = u_{1} + at_{1}\mspace{6mu},\mspace{6mu} u_{3} = u_{1} + a(t_{1} + t_{2})$

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

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