First slide

Collisions

Question

A particle of mass m moving in the x-direction with speed 2v is hit by another particle of mass 2m moving in the y-direction with speed v. If the collision is perfectly inelastic, the percentage loss in the Energy during the collision is close to

Moderate

A

56 %
B

62 %
C

44 %
D

50 %

Solution

$a p p l y l a w o f c o n s e r v a t i o n o f l i n e a r m o m e n t u m s u m o f m o m e n t u m o f b o t h b o d i e s b e f o r e c o l l i s i o n = s u m o f m o m e n t u m o f b o t h b o d i e s a f t e r c o l l i s i o n m 2 v \cdot \overset{\land}{i} + 2 m v \cdot \overset{\land}{j} = (m + 2 m) \cdot \bar{v}' h e r e \bar{v}' i f f i n a l v e l o c i t y o f b o t h b o d i e s a s t h e y s t i c k t o g e t h e r$
$∴ \bar{v}' = \frac{2 v}{3} (\overset{\land}{i} + \overset{\land}{j})$
$E n e r g y f i n a l = E_{f i n a l} = \frac{1}{2} (3 m) v'^{2} = \frac{3 m}{2} {(\frac{2 \sqrt{2}}{3} v)}^{2} = \frac{4}{3} m v^{2}$
$∴$ $p e r c e n t a g e l o s s o f e n e r g y = \frac{E_{i n i t i a l} - E_{f i n a l}}{E_{i n i t i a l}} \times 100$
$p e r c e n t a g e l o s s o f e n e r g y = \frac{3 m v^{2} - \frac{4}{3} m v^{2}}{3 m v^{2}} \times 100 p e r c e n t a g e l o s s o f e n e r g y = \frac{5}{9} \times 100 p e r c e n t a g e l o s s o f e n e r g y ≃ 56 %$

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