  In this paper we compute the dimension of all the higher secant varieties to
the Segre-Veronese embedding of $\mathbb{P}^n\times \mathbb{P}^1$ via the
section of the sheaf $\mathcal{O}(a,b)$ for any $n,a,b\in \mathbb{Z}^+$. We
relate this result to the Grassmann Defectivity of Veronese varieties and we
classify all the Grassmann $(1,s-1)$-defective Veronese varieties.
