Corrected entry: This episode breaks the laws of physics. After Kirk has been hyper-accelerated to Scalosian rate, he fires a phaser at Deela but the phaser light crawls out slowly and she steps out of the way. Phasers would fire at the speed of light, which is CONSTANT, a fundamental basis of modern physics. She would not have been able to avoid getting shot regardless of being hyper-accelerated.
Corrected entry: The physics here don't make any sense. If the Scalosians are moving 840 times faster, that means when they walk down a corridor (at 2 miles/hour, their speed) they are really moving at about 1680 miles per hour - that's Mach 2. The Scalosians don't create sonic booms, and they don't have any trouble stopping and turning corners despite the Mach 2 speeds they're moving at.