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A wooden block weighing 80 N rests on a rough horizontal plane, the coefficient of friction between the two being µ=0.4. If a bullet weighing 0.4 N is fired horizontally in to the block with muzzle velocity v= 500 m/s, how far will the block be displaced from its initial position? Assume that the bullet remains inside the block.
Asked by theavengers0203 | 24 May, 2023, 08:40: AM
answered-by-expert Expert Answer
Weight of bullet , ( m g  ) = 0.4 N ;
 
Velocity of bullet before impact, v = 500 m/s ;
 
Weight of wooden block , ( M g ) = 80 N ;
 
Let velocity of wooden block and bullet after impact = V
 
By conservation of momentum , total momentum of bullet embedded block after impact equals to
the initial momentum of bullet ( wooden block is under rest ) .
 
hence we have, 
 
begin mathsize 14px style m space v space equals space left parenthesis space M space plus space m space right parenthesis space V end style
 
begin mathsize 14px style left parenthesis m space g space right parenthesis space v space equals space left square bracket space left parenthesis space M space plus space m space right parenthesis g space right square bracket space V end style
 
begin mathsize 14px style V space equals space fraction numerator left parenthesis m g right parenthesis space v over denominator left parenthesis M plus m right parenthesis g end fraction space equals space fraction numerator 0.4 space cross times 500 over denominator 80.4 end fraction space m divided by s space equals space 2.49 space m divided by s end style
 
Kinetic energy K of bullet embedded wooden block  after impact is
 
begin mathsize 14px style K space equals space 1 half space left parenthesis M plus m right parenthesis space V squared end style
Work done W against friction force is
 
 begin mathsize 14px style W space equals space f o r c e space cross times space d i s tan c e space equals space left parenthesis space mu space left parenthesis M plus m right parenthesis space g space right parenthesis space cross times d end style
where μ is friction coefficient and d is the distance moved by bullet embedded wooden block  after impact.
 
By work-energy theorem , K = W 
 
begin mathsize 14px style mu space left parenthesis M plus m right parenthesis space g space d space equals space 1 half left parenthesis M plus m right parenthesis space V squared end style
begin mathsize 14px style d space equals space fraction numerator V squared over denominator 2 space mu space g space end fraction space equals space fraction numerator 2.49 space cross times space 2.49 over denominator 2 space cross times space 0.4 space cross times space 9.8 end fraction equals 0.79 space m end style
Answered by Thiyagarajan K | 24 May, 2023, 10:37: PM
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