Let $A_*$ and $B_*$ be projective chain complexes which are zero in dimensions $i=-1$ and down. Now suppose there exist maps $A_* \xrightarrow i B_* \xrightarrow r A_*$ such that $r \circ i = \text{id}_A$ and such that $i$ is a cofibration and $r$ is a fibration using the projective model structure.
https://ncatlab.org/nlab/show/model+structure+on+chain+complexes#CochainNonNeg
What can we say about the homology of $A_*$ and $B_*$? Are they isomorphic? If not, can we at least say that if $A_*$ is acyclic then $B_*$ is aswell? What bout the other way around, if $B_*$ is acyclic is $A_*$ acyclic aswell?
Such maps $i,r$ imply (in any abelian category) that $B_*\cong A_*\oplus \ker(r)$ so that the homology of $B_*$ is the direct sum of that of $A_*$ and that of $\ker(r)$.
In particular if $B_*$ is acyclic, so is $A_*$, but you can't say much more ($\ker(r)$ can be as weird as you want) : this has nothing to do with model structures or projectives.