Question

A bullet fired into a fixed target loses half of its velocity after penetrating 3 cm. How much further it will penetrate before coming to rest assuming that it faces constant resistance to motion​

Answer 1

${\mathfrak{\underline{\orange{\:\:\: Given:-\:\:\:}}}}$

$\:\:\:\:\bullet\:\:\:\sf{First \: penetrating \: length\:(s_{1}) = 3 \: cm}$

${\mathfrak{\underline{\pink{\:\:\:To \:Find:-\:\:\:}}}}$

$\:\:\:\:\bullet\:\:\:\sf{Left \: Penetration \: length \: before \: it \: comes \: to \: rest \:( s_{2} )}$

${\mathfrak{\underline{\blue{\:\:\: Calculation:-\:\:\:}}}}$

$\:\:\:\:\bullet\:\:\:\sf{Let \: Initial \: velocity = v\:m/s}$

$\:\:\:\:\bullet\:\:\:\sf{Left \: velocity \: after \: s_{1} \: penetration = \dfrac{v}{2} \:m/s}$

$\:\:\:\:\bullet\:\:\:\sf{s_{1} = \dfrac{3}{100} = 0.03 \: m}$

☯ As we know that,

$\dashrightarrow\:\: \sf{ {v}^{2} = {u}^{2} + 2as }$

$\dashrightarrow\:\: \sf{ \bigg(\dfrac{v}{2} \bigg)^{2} = {v}^{2} + 2a s_{1}}$

$\dashrightarrow\:\: \sf{ \dfrac{ {v}^{2} }{4} = {v}^{2} + 2 \times a \times 0.03 }$

$\dashrightarrow\:\: \sf{ \dfrac{ {v}^{2} }{4} - {v}^{2} = 0.06 \times a }$

$\dashrightarrow\:\: \sf{\dfrac{ - 3{v}^{2} }{4} = 0.06 \times a }$

$\dashrightarrow\:\: \sf{a = \dfrac{ - 3 {v}^{2} }{4 \times 0.06} }$

$\dashrightarrow\:\: \sf{ a = \dfrac{ - 25 {v}^{2} }{2}\:m/s^{2} ......(1) }$

$\:\:\:\:\bullet\:\:\:\sf{ Initial\:velocity=v\:m/s}$

$\:\:\:\:\bullet\:\:\:\sf{ Final \: velocity = 0 \: m/s }$

$\dashrightarrow\:\: \sf{ {v}^{2} = {u}^{2} + 2as}$

$\dashrightarrow\:\: \sf{{0}^{2} = {v}^{2} + 2 \times \dfrac{ - 25 {v}^{2} }{2} \times s }$

$\dashrightarrow\:\: \sf{ - {v}^{2} = - 25 {v}^{2} \times s }$

$\dashrightarrow\:\: \sf{ s = \dfrac{ - {v}^{2} }{ - 25 {v}^{2} }}$

$\dashrightarrow\:\: \sf{ s = \dfrac{1}{25} }$

$\dashrightarrow\:\: \sf{ s = 0.04 \: m }$

☯ For left penetration (s₂)

$\dashrightarrow\:\: \sf{s = s_{1} + s_{2} }$

$\dashrightarrow\:\: \sf{ 0.04 = 0.03 + s_{2}}$

$\dashrightarrow\:\: \sf{ s_{2} = 0.04 - 0.03 }$

$\dashrightarrow\:\: \sf{s_{2} = 0.01 \: m = {\boxed{\sf{\green{1 \: cm }}} }}$

$\star\:\sf{Left \: penetration \: before \: it \: come \: to \: rest \: is \:{\bf{ 1 \: cm}}}$