kernel of morphism of vector bundles

Question

Let $V$ and $W$ be two smooth vector bundles over a manifold $M$. Let $g^V_{ij}$ and $g^W_{ij}$ be the transition functions for $V$ and $W$ respectively. Let $f:V \rightarrow W$ be a morphism of vector bundles such that $f_p:V_p \rightarrow W_p$ has constant rank for all $p \in M$. Finally we denote by $\text{ker}f$ the kernel of the morphism $f$.

1. Then is is true that $\text{ker}f$ is a sub-bundle of $V$? If not, would the answer change if we assume $f_p$ is surjective instead?

2. If $\text{ker}f$ is a bundle in either of the above two cases then what would be the transition functions of $\text{ker}f$ in terms of $g^V_{ij}$, $g^W_{ij}$ and $f$ ?