Answer
$h = 1.2 ~t^2 ~m$
Work Step by Step
Let $t$ be the total time in the air. Then $\frac{t}{2}$ is the time to reach maximum height.
We know that:
$v_{y0} = \frac{gt}{2}$
We can use this to find an expression for maximum height $h$:
$h = v_{y0}~(\frac{t}{2}) + \frac{1}{2}a(\frac{t}{2})^2$
$h = (\frac{(9.80 ~m/s^2)(t)}{2})~(\frac{t}{2}) + \frac{(-9.80 ~m/s^2)}{2}(\frac{t}{2})^2$
$h = (\frac{9.80 ~m/s^2}{4})~t^2 - \frac{(9.80 ~m/s^2)}{8}~t^2$
$h = \frac{(9.80 ~m/s^2)}{8}~t^2 \approx 1.2 ~t^2 ~m$
Therefore, when we know the total time in the air $t$, the maximum height h is: $h = 1.2 ~t^2 ~m$
