Bullet hits a block, penetration depth and velocity

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 [itex]F_{ret}[/itex] 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 [itex]F_{ret}[/itex].

2. Relevant equations


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

[itex]\frac{1}{2}mv^{2}=F_{ret}l+\frac{1}{2}(m+M)V^{2}[/itex] (Eq 2)

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

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

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

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

Not sure if it is correct. Thanks for the help!

Simon Bridge

What is it that makes you doubt your answer?

rude man

Certainly your momentum and k.e. conservation equations are correct.