A bullet of mass $0.1\,kg$ moving horizontally with speed $400\,ms ^{-1}$ hits a wooden block of mass $3.9\,kg$ kept on a horizontal rough surface. The bullet gets embedded into the block and moves $20\,m$ before coming to rest. The coefficient of friction between the block and the surface is $........$ $\left(\text { Given } g=10 \,ms ^2\right. \text { ) }$
JEE MAIN 2023, Difficult
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$P _{ i }= P _{ f }(\text { Collision })$
$\Rightarrow(0.1)(400)=(0.1+3.9) v$
$\Rightarrow v =\frac{0.1 \times 400}{4}=10\,m / s$
$a =\frac{\mu mg }{ m }=\mu g$
Apply equation of motion,
$v ^2=u^2+2 as$
$\Rightarrow 0=(10)^2-2 \mu g \times 20$
$\Rightarrow 40 \mu g =100$
$\Rightarrow \mu=\frac{100}{2 \times 10 \times 20}=\frac{1}{4}$
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