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Velocity

A particle mo...

Question

A particle moving with uniform acceleration has average velocities $v_{1}$ , $v_{2}$ and $v_{3}$ over the successive intervals of time $t_{1}$ , $t_{2}$ adn $t_{3}$ respectively. teh value of $\frac{(v_{1} - v_{2})}{(v_{2} - v_{3})}$ will be

\frac{(t_{1} - t_{2})}{(t_{2} - t_{3})}

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\frac{(t_{1} - t_{2})}{(t_{2} + t_{3})}

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\frac{(t_{1} + t_{2})}{(t_{2} - t_{3})}

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\frac{(t_{1} + t_{2})}{(t_{2} + t_{3})}

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Solution

The correct option is D $\frac{(t_{1} + t_{2})}{(t_{2} + t_{3})}$
Assuming the particles moves the separation AB, BC and CD in time$ t_1 , t_2 and t_3 individually.
$S p e e d a t B = u + a t_{1},$
$S p e e d a t C = u + a (t_{1} + t_{2}),$
$S p e e d a t D = u + a (t_{1} + t_{2} + t_{3})$
Average velocity $(v_{1}) = (s t a r t i n g v e l o c i t y + f i n a l v e l o c i t y) / 2$
= $(u + u + a t_{1}) / 2$
= $(u + a t_{1} / 2)$ ................(1)
$v_{2} = u + a t_{1} + a t_{2} / 2$ .........................(2)
$v_{3} = u + a t_{1} + a t_{2} + a t_{3} / 2$ ............................(3)
Solving equation (1), (2) and (3),
$\frac{v_{1} - v_{2}}{v_{2} - v_{3}} = \frac{t_{1} + t_{2}}{t_{2} + t_{3}}$

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