Why is water expelled more powerfully from a trigger hose the moment its triggered? I think its something to do with acceleration but can't put my finger on it. 
Something about the most power needed as the greatest acceleration, as water flows up to full rate the accel and thus power decreases. Does that make sense?
 A: (Garden) hoses are elastic and they expand somewhat while being under pressure. The pressure in a hose is greatest while there is no flow, yet. As soon as you start spraying with the hose, the water pressure decreases and the elasticity of the material expels the extra stored volume in the hose. That's the short pulse of higher pressure water at the beginning.
There are two interesting limits to this effect. A hose made from the same material along its entire length will expand equally under the same pressure, but the water in the compressed volume will not be able to exit all at the same time. 
The first limit is given by the speed of sound in the water. The fastest possible velocity at which the hose can "detect" that one of its ends were opened is the speed of sound in the water. 
The second effect is that the hose has a resistance to flow which increases with length, i.e. the liquid near the nozzle will see a lower resistance and escape at a much higher velocity than the liquid volumes in the middle section and at the other end of the hose, which have to flow trough an ever longer piece of the hose against ever increasing resistance. That's why the pulse on a water hose will be strong, at first, and then gradually decline as the water has to flow trough ever longer lengths before it can exit. 
Curiously, one can observe the electrical equivalent of the pulse effect really nicely on electrical transmission lines, which (in some sense) work somewhat similarly to a water hose. Closing a load switch on one end of a charged (but initially open) electrical transmission line will cause a voltage pulse similar to the rapid pulse of water that you can observe on the hose. The electromagnetic wave inside the transmission line will flow at near the speed of light (typically at approx. 2/3 c) and it will make a nice square pulse that depends on the physical length of the transmission line. This is called a "cable pulser" or Pulse Forming Circuit and it's a widely used design in physics and electrical engineering. 
