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Let $A \otimes _tC$ be a {\em twisted tensor product} of an algebra $A$ and a coalgebra $C$, along a twisting cochain $t:C \rightarrow A$. By means of what is called the tensor trick and under some nice conditions, Gugenheim, Lambe and Stasheff proved in the early 90s that $A \ot _tC$ is homology equivalent to the objects $M \ot _{t'}C$ and $A \ot _{t''}N$, where $M$ and $N$ are strong deformation retracts of $A$ and $C$, respectively. In this paper, we attack this problem from the point of view of contractions. We find explicit contractions from $A\ot _t C$ to $M \ot _{t'}C$ and $A\ot_{t''}N$. Applications to the comparison of resolutions which split off of the bar resolution, as well as to some homological models for central extensions are given.