## Bullet hits a block, penetration depth and velocity

1. The problem statement, all variables and given/known data
A bullet with mass and speed v hits a wooden block of mass M that is situated at rest on a frictionless surface. It penetrates the block and gets trapped inside it as a result of a constant retardation force $F_{ret}$ that opposes relative motion between the two objects. Find the common speed of the bullet and the block V, and the penetration length l in terms of m, M, v, and $F_{ret}$.

2. Relevant equations

mv=(m+M)V (Eq 1)

$\frac{1}{2}mv^{2}=F_{ret}l+\frac{1}{2}(m+M)V^{2}$ (Eq 2)

3. The attempt at a solution
The common speed of m and M is $V=\frac{mv}{m+M}$ (Eq 3) via conservation of momentum.

$\frac{1}{2}mv^{2}=F_{ret}l+\frac{1}{2}\frac{m^{2}v^{2}}{(m+M)}$ (where I substituted Eq 3 into Eq 2)

Let's rearrange:
$F_{ret}l=\frac{1}{2}m\left( 1-\frac{m}{m+M}\right) v^{2}$

No let's solve for the penetration depth l:
$l=\frac{m\left(1-\frac{m}{m+M}\right) v^{2}}{2F_{ret}}$

Not sure if it is correct. Thanks for the help!
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 Quote by AbigailM 1. The problem statement, all variables and given/known data A bullet with mass and speed v hits a wooden block of mass M that is situated at rest on a frictionless surface. It penetrates the block and gets trapped inside it as a result of a constant retardation force $F_{ret}$ that opposes relative motion between the two objects. Find the common speed of the bullet and the block V, and the penetration length l in terms of m, M, v, and $F_{ret}$. 2. Relevant equations mv=(m+M)V (Eq 1) $\frac{1}{2}mv^{2}=F_{ret}l+\frac{1}{2}(m+M)V^{2}$ (Eq 2) 3. The attempt at a solution The common speed of m and M is $V=\frac{mv}{m+M}$ (Eq 3) via conservation of momentum. $\frac{1}{2}mv^{2}=F_{ret}l+\frac{1}{2}\frac{m^{2}v^{2}}{(m+M)}$ (where I substituted Eq 3 into Eq 2) Let's rearrange: $F_{ret}l=\frac{1}{2}m\left( 1-\frac{m}{m+M}\right) v^{2}$ No let's solve for the penetration depth l: $l=\frac{m\left(1-\frac{m}{m+M}\right) v^{2}}{2F_{ret}}$ Not sure if it is correct. Thanks for the help!
Certainly your momentum and k.e. conservation equations are correct.