# A bullet through a wall

A bullet with an initial velocity of v0, temperature t0(melting temperature), goes through a wall, and its velocity reduces to v. You know Lb(latent specific heat). You are asked to derive the ratio between the final mass of the bulet and the initial one.Oh, and you know that the energy lost by the bullet it's transformed in a ratio f(<1), to heat.
Here's what i have done:
We know that F=dp/dt, implies F=dm/dt*v + dv/dt*m;
We also know that dL=-fdQ, dQ=-dm*Lb(because dm is negative), which leads to dL=f*dm*Lb, dL=F*dx=F*v*dt=(dm/dt*v + dv/dt*m)*v*dt;
dL=dm*v^2 + m*v*dv
i earlier stated dL=f*dm*Lb, combining the last two gives:
dm*v^2 + m*v*dv = f*dm*Lb
m*v*dv=dm(f*Lb-v^2)
(v*dv)/(f*Lb-v^2)=dm/m
u=f*Lb-v^2 => du=-2v*dv => v*dv=-du/2
-1/2 * (du/u)=dm/m
=> -1/2 * ln (u2/u1) = ln (mf/mi)
(u2/u1)^(-1/2)=mf/mi
sqrt(u1/u2)=mf/mi
sqrt((f*Lb-v0^2)/(f*Lb-v^2))=mf/mi
the problem here is like this: the "-" sign demands that v0 cannot be greater than sqrt(f*Lb) which is illogical. I don't know what i have missed.

Did you not understand what i said?

Andrew Mason