According to Wikipedia, "Hang time [...] in basketball [is] the length of time a player stays in the air after jumping to score or pass the ball."
Therefore, it means that 1.09 seconds passed between the moment when the player jumped and landed.
It means that half of time the player was moving upward in the air until reached the maximum height and half of time he was moving down. Of course, we assume that air resistance is 0.
That is why the maximum height requires you to use (1.09/2) seconds.
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