Q. A particle moves with constant acceleration and $v_{1},v_{2}$ and $v_{3}$ denote the average velocities in the three successive interval $t_{1},\ t_{2}$ and $t_{3}$ of time. Which of the following relations is correct?
A) $\frac{v_{1} - v_{2}}{v_{2} - v_{3}} = \frac{t_{1} - t_{2}}{t_{2} + t_{3}}$
B) $\frac{v_{1} - v_{2}}{v_{2} - v_{3}} = \frac{t_{1} - t_{2}}{t_{1} - t_{3}}$
C) $\frac{v_{1} - v_{2}}{v_{2} - v_{3}} = \frac{t_{1} - t_{2}}{t_{2} - t_{3}}$
D) $\frac{v_{1} - v_{2}}{v_{2} - v_{3}} = \frac{t_{1} + t_{2}}{t_{2} + t_{3}}$
Explanation
(d) $v_{1} = \frac{u + v_{1}^{'}}{2} = \frac{u + u + at_{1}}{2} = u + \frac{1}{2}at_{1}$
$v_{2} = \frac{v_{1}^{'} + v_{2}^{'}}{2} = \frac{\left( u + at_{1} \right) + u + a(t_{1} + t_{2})}{2}$
$= u + at_{1} + \frac{1}{2}at_{2}$
$v_{3} = \frac{v_{2}^{'} + v_{3}^{'}}{2} = \frac{\left( u + at_{1} + at_{2} \right) + u + a(t_{1} + t_{2} + t_{3})}{2}$
$= u + at_{1} + at_{2} + \frac{1}{2}at_{3}$
Then, $v_{1} - v_{2} = - \frac{1}{2}a(t_{1} + t_{2})$
$v_{2} - v_{3} = - \frac{1}{2}a(t_{2} + t_{3})$
$\therefore\ \frac{v_{1} - v_{2}}{v_{2} - v_{3}} = \frac{t_{1} + t_{2}}{t_{2} + t_{3}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.