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{\purple{\:\:\: Given:-\:\:\:}}}} \\ \\`

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

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`{\mathfrak{\underline{\purple{\:\:\:To \:Find:-\:\:\:}}}} \\ \\`

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

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`{\mathfrak{\underline{\purple{\:\:\: Calculation:-\:\:\:}}}} \\ \\`

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

`\:\:\:\:\bullet\:\:\:\sf{Left \: velocity \: after \: s_(1) \: penetration = (v)/(2) \:m/s} \\\\`

`\:\:\:\:\bullet\:\:\:\sf{s_(1) = (3)/(100) = 0.03 \: m}`

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☯ As we know that,

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`\dashrightarrow\:\: \sf{ {v}^(2) = {u}^(2) + 2as }`

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

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`\dashrightarrow\:\: \sf{ \frac{ {v}^(2) }{4} = {v}^(2) + 2 * a * 0.03 }`

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`\dashrightarrow\:\: \sf{ \frac{ {v}^(2) }{4} - {v}^(2) = 0.06 * a }`

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`\dashrightarrow\:\: \sf{\frac{ - 3{v}^(2) }{4} = 0.06 * a }`

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`\dashrightarrow\:\: \sf{a = \frac{ - 3 {v}^(2) }{4 * 0.06} }`

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

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

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

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`\dashrightarrow\:\: \sf{ {v}^(2) = {u}^(2) + 2as}`

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`\dashrightarrow\:\: \sf{{0}^(2) = {v}^(2) + 2 * \frac{ - 25 {v}^(2) }{2} * s }`

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`\dashrightarrow\:\: \sf{ - {v}^(2) = - 25 {v}^(2) * s }`

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`\dashrightarrow\:\: \sf{ s = \frac{ - {v}^(2) }{ - 25 {v}^(2) }}`

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`\dashrightarrow\:\: \sf{ s = (1)/(25) }`

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

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☯ For left penetration (s₂)

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`\dashrightarrow\:\: \sf{s = s_(1) + s_(2) }`

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`\dashrightarrow\:\: \sf{ 0.04 = 0.03 + s_(2)}`

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`\dashrightarrow\:\: \sf{ s_(2) = 0.04 - 0.03 }`

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`\dashrightarrow\:\: \sf{s_(2) = 0.01 \: m = {\boxed{\sf{\purple{1 \: cm }}} }}`

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