First slide

Rectilinear Motion

Question

A particle moving in a straight line covers half the distance with Speed $V_{o}$ .The other half of the distance is covered in two equal time intervals with speed $V_{1}$ and $V_{2}$ respectively. The average speed of the particle during this motion is

Difficult

A

$\frac{V_{o} (V_{1} + V_{2})}{V_{o} + V_{1} + V_{2}}$
B

$\frac{2 V_{o} (V_{1} + V_{2})}{V_{o} + V_{1} + V_{2}}$
C

$\frac{2 V_{o} (V_{1} + V_{2})}{2 V_{o} + V_{1} + V_{2}}$
D

$\frac{V_{o} (V_{1} + V_{2})}{2 V_{o} + V_{1} + V_{2}}$

Solution

$A t o B : \to g i v e n v e l o c i t y = V_{0}; f o r d i s \tan c e = \frac{d}{2} t i m e t = \frac{d i s \tan c e}{v e l o c i t y} t = \frac{(\frac{d}{2})}{V_{0}} = \frac{d}{2 V_{0}} ---(1)$
$B t o D : \to i f t i m e t a k e n f o r n e x t d i s \tan c e = \frac{d}{2} l e t t i m e t a k e n f o r \frac{d}{2} be t_{0} f o r a t i m e o f \frac{t_{0}}{2} v e l o c i t y = V_{1}; a n d r e m a i n i g t i m e o f \frac{t_{0}}{2} v e l o c i t y = V_{2} f o r d i s \tan B D \to \frac{d}{2} = V_{1} {\frac{t_{0}}{2}}_{} + V_{2} {\frac{t_{0}}{2}}_{}$
$t_{0} = \frac{d}{(V_{1} + V_{2})}$ ----(2)
$A v e r a g e S p e e d = \frac{t o t a l d i s \tan c e}{t o t a l t i m e} A v e r a g e S p e e d = \frac{d}{t + t_{0}} s u b s t i t u t e e q n (1) a n d (2) A v e r a g e S p e e d = \frac{2 V_{0} (V_{1} + V_{2})}{2 V_{0} + (V_{1} + V_{2})}$

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