From Mechanics

The ball hits the ground when the \displaystyle \mathbf{j} component of the position vector is zero.

We calculate the position vector using \displaystyle \mathbf{r}=\mathbf{u}t+\frac{1}{2}\mathbf{a}{{t}^{\ 2}}+{{\mathbf{r}}_{0}} .

Here \displaystyle \mathbf{u}=8\mathbf{i}+10\mathbf{j}\text{ m}{{\text{s}}^{\text{-1}}}, \displaystyle \ \mathbf{a}=-10\mathbf{j}\ \text{ m}{{\text{s}}^{\text{-2}}} and \displaystyle \ \mathbf{r}_{0}=0 giving,

\displaystyle \mathbf{r}=(8\mathbf{i}+10\mathbf{j})t+\frac{1}{2}(-10\mathbf{j}){{t}^{\ 2}}

The \displaystyle \mathbf{j} component of this position vector is \displaystyle 10t-5{{t}^{\ 2}}.

This is zero if \displaystyle t=2\ \text{s}.

The other solution \displaystyle t=0 is the trivial solution restating the fact that the ball is at the origin at the start.