Wouldn't it be better to put the output of a ac near the floor of a building rather than the top?

I'm thinking that it would be better to design a ac to have the outputting cold air at the bottom since hot air rises and as it rises it gets exhausted by a fan. I thought of this since at gymnasium at my school got a new ac system and the exhaust and output seems to be on the same level (near the ceiling)

If anyone can tell me why this doesn't work or why it isn't used if it does work

It would be appreciated!