Let $M$ be a smooth manifold and $\xi:E\to M$ a real vector bundle over $M$. Suppose we fix a metric $g$ on $E$ so that we can define the unit sphere bundle $S(E)\to M$.
How are the characteristic classes of the tangent bundle $TS(E)$ related to the characteristic classes of the vector bundle $\xi$? For example, can we compute $w_2(\xi)$ and $p_1(\xi)$ in terms of the characteristic classes of $TS(E)$?
I think the choice of metric on $E$ is irrelevant here since the space of metrics in contractible. Maybe we could use the Leray-Hirsch theorem?