That was funny considering both the transporter room and sickbay are on lower decks.
I'm not so sure about the transporter room. In Star Trek 2009
, could Chekov made it there while Kirk and Sulu fell if he had to stop and take a turbolift?
Yes. Spock went from the shuttlebay to the bridge in a matter of seconds. The Abramsprise's turbolifts make modern people movers weep in envy.