That is an interesting problem. There are a few variables in play: 1. the time to open the hatch 2. the time to pop-out the phaser emitter 3. the time to point the emitter at the target An external sliding hatch or folding out hatch on average takes 1.2 seconds to open or close (see "Galileo Seven" shuttlecraft door open and close and "The Mark of Gideon" external window cover opening). The time to pop-out the phaser emitter might only take another second or two and it could rotate to aim at the target in the same motion. So maybe 2 to 3 seconds to popout and fire. Although "Balance of Terror" seems to suggest that phasers could be fired almost immediately ("phasers fire point-blank!") so maybe even faster than that. "Journey to Babel" would suggest that the phasers have a maximum traversal speed as they were unable to hit the Orion ship when there is a huge difference in relative speed and distance between enemy ships. Some possible alternative approaches: 1. phaser emitters fire behind energy transparent sections in the hull so they don't need time to wait for the hatch to open. This allows them to be able to lock-on (track) and then fire immediately. 2. phaser emitters wait for a hatch to open but track and fire immediately without popping up.