Time taken to reach height below initial point

Question

I am trying to solve the following. It is in my notes, I believe I need to solve:

$-H = v_0cos\alpha*t\bf{j}$$+(v_0sin\alpha*t-gt^2/2)\bf{k}$

as a quadratic equation? If so, how do I solve a component-wise quadratic?

Thanks

Answer 1

Since $\bf k$ is the vertical vector, to find the time of any altitude $h$, you solve for the time the coefficient of $\bf k$ is $h$, so you would solve $h=v_0t\sin \alpha -gt^2/2$ The part involving $\bf j$ doesn't contribute. Then if you want to find the horizontal position you can take the $t$ you got and plug it in.