A while back I did some similar calculations. I arbitrarily chose the horizontal deceleration of the arrow to be 0.25 ft/s per yard travelled. And I dragged my calcs out to 50 yds, at which it takes an arrow starting at 300 fps, .51 sec to get there.
The speed of sound is 1116 ft/s. It takes .134 sec to go 50 yds. Difference in time is .376.
At 30 yds an arrow takes .294 sec & sound takes .08 sec. Difference of .214 sec.

A deer does not have less time to react the farther away it is.