Let $I$ be an interval and $f \colon I \to \mathbb{R}$ is a positive function such that $M = I \times_f S^n(1)$ is the warped product Riemannian manifold, which is a hypersurface in $\mathbb{R}^{n+2}$. Here $I$ denotes an interval and $S^n(1)$ is the unit sphere, both equipped with their standard metric. I'm now asked to determine the normal vector field of $M$. The normal vector field is of course the vector field that is normal to $T_pM$. But $T_pM = T_pI \oplus T_pS^n(1)$ so I think that it is enough to find a normal vector field on $I$ and $S^n(1)$. The normal vector field on $S^n(1)$ is of course just the position vector but I'm not sure what the normal vector on the interval is, nor if my approach is correct.
EDIT: the immersion is given by
$\varphi : I \times S^n \to \mathbb{R}^{n+2} : (t, y_0,...,y_n) \mapsto \big(\alpha_1(t), \alpha_2(t)y_0,..., \alpha_2(t)y_n\big) $
where $\alpha \colon I \to \mathbb{R}^{2}$ is an arc lenght parametrised curve with $\alpha_2(t) > 0$.