physics

mechanics

Easy

Question

$text Two balls have masses in ratio 2:1. Their velocities are in the ratio of end text 1 colon 2 text . end text T h e space text coefficient of restitution is end text 5 over 6 text . Then the final velocity of each ball is end text$

$\frac{29}{18} & \frac{7}{18} of its initial velocity$
$\frac{6}{5} & \frac{5}{6} of its initial velocity$
$\frac{36}{25} & \frac{25}{36} of its initial velocity$
$\frac{25}{36} & \frac{36}{25} of its initial velocity$

The correct answer is: $\frac{29}{18} & \frac{7}{18} of its initial velocity$

$text Given end text m subscript 1 equals 2 m subscript 2 semicolon u subscript 2 equals 2 u subscript 1 semicolon fraction numerator v subscript 2 minus v subscript 1 over denominator u subscript 1 minus u subscript 2 end fraction equals 5 over 6 text According to law of conservation of linear momentum, end text m subscript 1 u subscript 1 plus m subscript 2 u subscript 2 equals m subscript 1 v subscript 1 plus m subscript 2 v subscript 2 text or end text 2 m subscript 2 u subscript 1 plus m subscript 2 2 u subscript 1 equals 2 m subscript 2 v subscript 1 plus m subscript 2 v subscript 2 not stretchy rightwards double arrow 2 v subscript 1 plus v subscript 2 equals 4 u subscript 1 not stretchy rightwards arrow left parenthesis 1 right parenthesis fraction numerator v subscript 2 minus v subscript 1 over denominator u subscript 1 minus u subscript 2 end fraction equals 5 over 6 not stretchy rightwards double arrow 6 v subscript 2 minus 6 v subscript 1 equals negative 5 u subscript 1 not stretchy rightwards arrow left parenthesis 2 right parenthesis table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em rowspacing 3 pt end attributes row cell text On solving equations end text left parenthesis 1 right parenthesis text and end text left parenthesis 2 right parenthesis comma v subscript 2 equals left parenthesis 7 divided by 9 right parenthesis u subscript 1 equals left parenthesis 7 divided by 9 right parenthesis u subscript 2 divided by 2 equals 7 divided by 18 u subscript 2 end cell row cell text Similarly, end text v subscript 1 equals left parenthesis 29 divided by 18 right parenthesis end cell row blank end table$

$T w o space p a r t i c l e s space o f space m a s s e s space m subscript 1 space a n d space m subscript 2 space a r e space c o n n e c t e d space b y space a space l i g h t space a n d space i n e x t e n s i b l e space s t r i n g space w h i c h space p a s s e s space o v e r space a space f i x e d space p u l l e y. space I n i t i a l l y comma space t h e space p a r t i c l e space m subscript 1 space m o v e s space w i t h space a space v e l o c i t y space v subscript 0 w h e n space t h e space s t r i n g space i s space n o t space t a u t. space N e g l e c t i n g space f r i c t i o n space i n space a l l space c o n t a c t i n g space s u r f a c e s comma space f i n d space t h e space v e l o c i t i e s space o f space t h e space p a r t i c l e s space m subscript 1 space a n d space m subscript 2 space j u s t space a f t e r space t h e space s t r i n g space i s space t a u t.$

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Mechanics

Physics

$A space p l a t e space o f space m a s s space M space i s space h e l d space a t space r e s t space b y space f i r i n g space b u l l e t s space f r o m space b e l o w. space E a c h space b u l l e t space h a s space a space m a s s space m space v e l o c i t y space u space left parenthesis u p right parenthesis space j u s t space b e f o r e space h i t t i n g space a n d space s t o p s space a f t e r space h i t t i n g space t h e space p l a t e space f o r space a space m o m e n t space a n d space f a l l s. space D e t e r m i n e space t h e space n u m b e r space o f space b u l l e t s space s t r i k i n g space t h e space p l a t e space p e r space u n i t space t i m e.$

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Mechanics

Physics

$A space b a l l space o f space m a s s space 1 space k g space i s space a t t a c h e d space t o space a n space i n e x t e n s i b l e space s t r i n g. space T h e space b a l l space i s space r e l e a s e d space f r o m space t h e space p o s i t i o n space s h o w n space i n space f i g u r e. space F i n d space t h e space i m p u l s e space i m p a r t e d space b y space t h e space s t r i n g space t o space t h e space b a l l space i m m e d i a t e l y space a f t e r space t h e space s t r i n g space b e c o m e s space t a u t.$

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Mechanics

Physics

$A space b o b space o f space m a s s space M space i s space s u s p e n d e d space b y space a n space i n e x t e n s i b l e space s t r i n g. space A space p a r t i c l e space o f space m a s s space m comma space m o v i n g space w i t h space v e l o c i t y space v subscript 0 space s t r i k e space t h e space b o b space a s space s h o w n space i n space t h e space f i g u r e. space I f space t h e space p a r t i c l e space c o m e s space t o space r e s t space j u s t space a f t e r space c o l l i s i o n comma space t h e n space c a l c u l a t e space text velocity of the bob just after collision. end text$

PhysicsMechanics

Mechanics

Physics

$A space b o b space o f space m a s s space M space i s space s u s p e n d e d space b y space a n space i n e x t e n s i b l e space s t r i n g. space A space p a r t i c l e space o f space m a s s space m comma m o v i n g space w i t h space v e l o c i t y space v subscript 0 space s t r i k e space t h e space b o b space a s space s h o w n space i n space t h e space f i g u r e. space I f space t h e space p a r t i c l e space c o m e s space t o space r e s t space j u s t space a f t e r space c o l l i s i o n comma space t h e n space c a l c u l a t e space i m p u l s e space o f space t e n s i o n space i n space t h e space s t r i n g$

PhysicsMechanics

Mechanics

Physics

$A space t i m e space v a r y i n g space f o r c e comma space F equals 2 space t space i s space a c t i n g space o n space a space p a r t i c l e space o f space m a s s space 2 space k g space m o v i n g space a l o n g x space minus a x i s. space v e l o c i t y space o f space t h e space p a r t i c l e space i s space 4 space m space divided by space s space a l o n g space n e g a t i v e space x space minus a x i s space a t space t i m e t equals 0 comma space F i n d space t h e space v e l o c i t y space o f space t h e space p a r t i c l e space a t space t h e space e n d space o f space 4 space s$

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Mechanics

Physics

$A space b a l l space o f space m a s s space 200 space g space i s space p r o j e c t e d space w i t h space a space v e l o c i t y space o f space 30 space m space divided by space s space a t space 30 to the power of 0 space f r o m space h o r i z o n t a l. space U sin g space t h e space c o n c e p t space o f space i m p u l s e comma space f i n d space c h a n g e space i n space v e l o c i t y space i n space 2 space s T a k e space g equals 10 space m space divided by space s space squared$

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Mechanics

Physics

$T o space w h a t space m a x i m u m space h e i g h t space w i l l space t h e space b l o c k space r i s e$

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Mechanics

Physics

$A space s m a l l space b l o c k space o f space m a s s space m space equals space 1 space k g space i s space p l a c e d space o v e r space a space w e d g e space o f space m a s s space M space equals space 4 space k g space a s space space s h o w n space i n space f i g u r e. space M a s s space m space i s space r e l e a s e d space f r o m space r e s t space A l l space s u r f a c e s space a r e space s m o o t h. space O r i g i n space O space i s space a s space s h o w n comma space left parenthesis g space equals space 10 space m divided by s squared right parenthesis F i n a l space v e l o c i t y space o f space t h e space w e d g e space i s.......... m divided by s$

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Mechanics

Physics

$bold italic P bold italic a bold italic s bold italic s bold italic a bold italic g bold italic e bold colon A space b l o c k space o f space m a s s space 1 space k g space i s space m o v i n g space t o w a r d s space a space m o v a b l e space w e d g e space o f space m a s s space 2 space k g space a s space s h o w n space i n space f i g u r e. space A l l space s u r f a c e s space a r e space s m o o t h. space W h e n space t h e space b l o c k space l e a v e s space t h e space w e d g e f r o m space t o p comma space i t s space v e l o c i t y space i s space m a k i n g space a n space a n g l e space theta space equals space 30 º space w i t h space h o r i z o n t a l. T h e space v a l u e space o f space v subscript 0 space i n space m divided by s space i s space$

PhysicsMechanics

Mechanics

Physics

$A space b a l l space o f space m a s s space m space i s space m o v i n g space t o w a r d s space a space b a t s m a n space a t space a space s p e e d space v. space T h e space b a t s m a n s t r i k e s space t h e space b a l l space a n d space d e f l e c t s space i t space b y space a n space a n g l e space theta space w i t h o u t space c h a n g i n g space i t s space s p e e d. space T h e i m p u l s e space i m p a r t e d space t o space t h e space b a l l space i s space g i v e n space b y space colon$

PhysicsMechanics

Mechanics

Physics

$A space b o d y space o f space m a s s space 5 space k g space i s space m o v i n g space a l o n g space a space s t r a i g h t space l i n e space w i t h space a space v e l o c i t y space o f space 2 space m s to the power of – 1 end exponent. space A space v a r i a b l e space f o r c e space F space s t a r t e d space a c t i n g space o n space i t space f o r space a space t i m e space p e r i o d space o f space 0.5 s space a s space s h o w n space i n space f o r c e minus t i m e c u r v e. space T h e space f i n a l space v e l o c i t y space o f space t h e space b l o c k space i s$

PhysicsMechanics

Mechanics

Physics

$T w o space p o i n t space m a s s e s space c o n n e c t e d space b y space a n space i d e a l space s t r i n g space a r e space p l a c e d space o n space a space s m o o t h space h o r i z o n t a l space s u r f a c e space a s space s h o w n space i n space t h e space d i a g r a m. space A space s h a r p space i m p u l s e space o f space 10 space k g minus m divided by s space i s space g i v e n space t o space t h e 5 space k g space m a s s. space T h e space v e l o c i t y space o f space t h e space 10 space k g space m a s s space w i l l space b e$

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Mechanics

Physics

$T h e space m a g n i t u d e space o f space t h e space f o r c e space left parenthesis i n space N right parenthesis space a c t i n g space o n space a space b o d y space v a r i e s space w i t h space t i m e space t space left parenthesis i n space m s right parenthesis space a s space s h o w n. space A B comma space B C space a n d space C D space a r e space s t r a i g h t space l i n e space s e g m e n t s. space T h e space m a g n i t u d e space o f space t h e space t o t a l space i m p u l s e space o f space t h e f o r c e space o n space t h e space b o d y space f r o m space t space equals space 4 space mu s space t o space t space equals space 16 space mu s space i s space$

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Mechanics

Physics

$A space g r a p h space b e t w e e n space k i n e t i c space e n e r g y space a n d space m o m e n t u m space o f space a space p a r t i c l e space i s space p l o t t e d space a s space s h o w n space i n space t h e space f i g u r e. space T h e space m a s s space o f space t h e space m o v i n g space p a r t i c l e space i s space$

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