We know that if $X$ is a smooth connected variety over a field $k$, then any line bundle on $X\times_k\mathbb{A}^1$ is from a line bundle on $X$. This is simply because they have the same Picard group. This has a generalization, in fact, $X\times_k\mathbb{A}^1\to X$ induces an isomorphism of $CH_r(X)\to CH_{r+1}(X\times_k\mathbb{A}^1)$.

But I am thinking about vector bundles on $X\times_k\mathbb{A}^1$, I would not believe that any vector bundle on $X\times_k\mathbb{A}^1$ is from $X$. Can anyone give me a counter example ? I think it might be easy to take $X=\mathbb{P^1}$ where the vector bundles are completely split (into line bundles), while the same should not be true for $X\times_k\mathbb{A}^1$. Can anyone give me an example when I take $X$ to be an Abelian variety? for example an elliptic curve.