physics

mechanics

Easy

Question

$A n space e x p l o s i o n space b l o w s space a space r o c k space i n t o space t h r e e space p a r t s. space T w o space p i e c e s space g o space o f f space a t space r i g h t space a n g l e s space t o space e a c h space o t h e r semicolon space 1.0 space k g space p i e c e space w i t h space a space v e l o c i t y space o f space 12 space m space divided by space s space a n d space o t h e r space 2.0 space k g space p i e c e space w i t h space a space v e l o c i t y space o f space 8 space m space divided by space s. space I f space t h e space t h i r d space p i e c e space f l i e s space o f f space w i t h space a space v e l o c i t y space o f space 40 space m space divided by space s comma space c o m p u t e space t h e space m a s s space o f space t h e space t h i r d space p i e c e.$

$0.5 k g$
$1 k g$
$0.4 k g$
$1.5 k g$

The correct answer is: $0.5 k g$

$text Let end text m subscript 1 comma m subscript 2 text and end text m subscript 3 text be the masses of the three pieces. end text m subscript 1 equals 1.0 k g comma m subscript 2 equals 2.0 k g text Let end text v subscript 1 equals 12 m divided by s comma v subscript 2 equals 8 m divided by s comma v subscript 3 equals 40 m divided by s text and end text v subscript 1 text and end text v subscript 2 space a n d space text directed along end text x text - and end text y text -axes, respectively, and end text v subscript 3 text be directed as shown end text$

$B y space t h e space p r i n c i p l e space o f space c o n s e r v a t i o n space o f space m o m e n t u m comma space i n i t i a l space m o m e n t u m space i s space z e r o. space H e n c e comma space table attributes columnalign left end attributes row cell text along end text x minus text axis end text colon 0 equals m subscript 1 v subscript 1 minus m subscript 3 v subscript 3 cos theta end cell row cell text along end text y minus text axis end text colon 0 equals m subscript 2 v subscript 2 minus m subscript 3 v subscript 3 cos theta end cell end table ⟹ m subscript 1 v subscript 1 equals m subscript 3 v subscript 3 C o s theta text and end text m subscript 2 v subscript 2 equals m subscript 3 v subscript 3 C o s theta m subscript 2 superscript 3 equals fraction numerator 1 squared left parenthesis 12 right parenthesis squared plus left parenthesis 2 right parenthesis squared left parenthesis 8 right parenthesis squared over denominator left parenthesis 40 right parenthesis squared end fraction ⟹ m plus 3 equals 0.5 k g$

$A space b o d y space o f space m a s s space 5 space m space e x p l o d e s space a t space r e s t space b r e a k space u p space i n t o space t h r e e space p a r t s. space I f space t w o space p a r t s space h a v i n g space e q u a l space m a s s e s space f l y space o f t space p e r p e n d i c u l a r l y space t o space e a c h space o t h e r space w i t h space a space v e l o c i t y space o f space 18 space m space divided by space s comma space t h e n space c a l c u l a t e space t h e space v e l o c i t y space o f space t h e space t h i r d space p a r t space w h i c h space h a s space a space m a s s space 3 space t i m e s space t h e space m a s s space o f space e q u a l space p a r t s.$

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Mechanics

Physics

$A space 5.0 space g space b u l l e t space m o v i n g space a t space 100 space m space divided by space s space s t r i k e s space a space log. space A s s u m e space t h a t space t h e space b u l l e t space u n d e r g o e s space u n i f o r space d e c e l e r a t i o n space a n d space s t o p s space i n space 6.0 space c m. space F i n d space t h e space t i m e space t a k e n space f o r space t h e space b u l l e t space t o space s t o p$

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Mechanics

Physics

$T h e space n e t space f o r c e space v e r s u s space t i m e space g r a p h space o f space a space r o c k e t space i s space s h o w n space i n space f i g u r e. space T h e space m a s s space o f space t h e space r o c k e t space i s space 1200 space k g space C a l c u l a t e space v e l o c i t y space o f space r o c k e t comma space 16 space s e c o n d s space a f t e r space s t a r t i n g space f r o m space r e s t. space N e g l e c t space g r a v i t y.$

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Mechanics

Physics

$A space b a l l space o f space m a s s space m comma space t r a v e l l i n g space w i t h space v e l o c i t y space 2 i with hat on top plus 3 j with rightwards arrow on top space j r e c e i v e s space a n space i m p u l s e minus 3 m i with hat on top. space W h a t space i s space t h e space v e l o c i t y d space t h e space b a l l space i m m e d i a t e l y space a f t e r w a r d s ?$

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Mechanics

Physics

$A space t r u c k space o f space m a s s space 2 cross times 10 cubed space k g space t r a v e l l i n g space a t space 4 space m space divided by space s space i s space b r o u g h t space t o space r e s t space i n space 2 space s space w h e n space i t space s t r i k e s space a space w a l l. space W h a t space f o r c e space left parenthesis a s s u m e space c o n s tan t right parenthesis space i s space e x e r t e d space b y space t h e space w a l l space ?$

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Mechanics

Physics

$A space b u l l e t space o f space m a s s space 10 to the power of negative 3 end exponent k g space s t r i k e s space a n space o b s t a c l e space a n d space m o v e s space a t space 6 space r space t o space i t s space o r i g i n a l space d i r e c t i o n. space I f space i t s space s p e e d space a l s o space c h a n g e s space f r o m space 20 space m space divided by space s space t o space 10 space m space divided by space s. space F i n d space t h e space m a g n i t u d e space o f space i m p u l s e space a c t i n g space o n space t h e space b u l l e t space m equals 10 to the power of negative 3 end exponent k g.$

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Mechanics

Physics

$A space p a r t i c l e space o f space m a s s space 2 space k g space i s space i n i t i a l l y space a t space r e s t. space A space f o r c e space s t a r t s space a c t i n g space o n space i t space i n space o n e space d i r e c t i o n space w h o s e space m a g n i t u d e space c h a n g e s space w i t h space t i m e. space T h e space f o r c e space t i m e space g r a p h space i s space s h o w n space i n space f i g u r e. space F i n d space t h e space v e l o c i t y space o f space t h e space p a r t i c l e space a t space t h e space e n d space o f space 10 space s.$

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Mechanics

Physics

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

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Mechanics

Physics

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

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Mechanics

Physics

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

PhysicsMechanics

Mechanics

Physics

$A space b a l l space o f space m a s s space m space i s space a t t a c h e d space t o space a n space i n e x t e n s i b l e space l i g h t space s t r i n g. space N o w space t h e space b a l l space i s space r e l e a s e d space f r o m space t h e space p o s i t i o n space i n space t h e space s e c o n d space f i g u r e. space S u b s e q u e n t l y space w h i l e space f a l l i n g comma space s t r i n g space b e c o m e s space t a u t. A t space t h a t space m o m e n t space i m p u l s e space i m p a r t e d space b y space t h e space s t r i n g space t o space t h e space b a l l space w o u l d space b e$

PhysicsMechanics

Mechanics

Physics

$A space b o b space o f space m a s s space 10 space m space i s space s u s p e n d e d space u sin g space a n space i d e a l space s t r i n g. space I n i t i a l l y space b o b space i s space a t space r e s t. space T w o space p a r t i c l e s space e a c h space o f space m a s s space m space s t r i k e space t h e space b o b space s i m u l tan e o u s l y space a s space s h o w n. space T h e s e space p a r t i c l e s space s t i c k space a f t e r space c o l l i s i o n. space T h e n space i m p u l s e space i n space t h e space s t r i n g space d u e space t o space t e n s i o n space d u e space t o space i m p a c t space i s$

PhysicsMechanics

Mechanics

Physics

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

PhysicsMechanics

Mechanics

Physics

$B l o c k s space A space a n d space B space h a v e space m a s s e s space 40 space k g space a n d space 60 space k g space r e s p e c t i v e l y. space T h e y space a r e space p l a c e d space o n space a space s m o o t h space s u r f a r space a n d space t h e space s p r i n g space c o n n e c t e d space b e t w e e n space t h e m space i s space s t r e t c h e d space b y space 1.5 space m. space I f space t h e y space a r e space r e l e a s e d space f r o m space r e s t space d e t e r m i n e space t h e space s p e e d s space o f space b o t h space b l o c k s space a t space t h e space i n s tan t space t h e space s p r i n g space b e c o m e s space u n s t r e t c h e d.$

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Mechanics

Physics

$F o r space t h e space f i g u r e space s h o w n comma space t h e space w e d g e space plus space p a r t i c l e space i s space r e s t i n g space a t space a space v e r t i c a l space w a l l. space T h e space p a r t i c l e o f space m a s s space m equals M over 2 space i s space r e l e a s e d space f r o m space r e s t space f r o m space t h e space p o s i t i o n space s h o w n. space t h e space w e d g e space h a s space m a s s M. space F i n d space t h e space m a x i m u m space h e i g h t space r e a c h e d space b y space t h e space p a r t i c l e space o n space t h e space c i r c u l a r space g r o o v e.$

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