A water rocket works "kind of" like a rocket at first, but it has a fixed amount of pressure water, and dissipates rapidly, without renewal of pressure.
In a solid fuel rocket, the fuel (actually fuel, oxidizer, and catalyst mixed) has holes drilled through it, and the surface area stays relatively constant as it burns, and it produces a relatively constant thrust force over the fuel burn duration. Not exactly constant, but relatively. So, the thrust stays relatively constant. Ideally, it would be a square funtion with a constant amplitude. in reality, it builds over time, reaches a peak, and decays slightly until the fuel begins to burn out, and then decays to zero over time. So the ideal square pulse has a slope upward at the front (steep), and slope downward (slowly) over the thrust duration, and slope (steep) downward when the fuel starts to lose surface area rapidly as it burns out.
The impulse momentum produced, which imparts kinetic energy to the rocket, is the integration of the force produced over time [usually gives pound-seconds [(slug-ft) / sec] or Newton-seconds (kg-m)/s]. Most load cells can give you a real-time readout of the force, in pounds or Newtons, and any decent oscilloscope (we used Nicolet scopes with digital math capability) can perform a digital integration of the force waveshape to get Newton-seconds or pound-seconds (which can be converted into Newton-seconds). Of course, the circuits have rise tme constraints, but the accuracy is very good, especially with piezoelettric presure transducers (we usually used Kistler piezo units and Honeywell-Sensotech load cells).
The thrusters we designed and built worked OK for the Space Shuttle. NASA agreed with my analyses. So did Martin Space Systems, and the Space Shuttle did some things wrong, but the thrust supplied by our thrusters in the nose caps of the SRB's and other thruster locations worked as desgned for a LONG TIME.
Sorry Engguy, but my stuff has flown in space for the USA and functioned as designed, perfectly. I didn't get an "x" in Physics; I got an "A," at least as far as my company was concerned. When you get something into space and it works for more than 30 years, come back and tell us about it. I was a test engineer and test manager, and was for more than 30 years in electronics and aerospace. I taught electronics and Physics, and still do on occasion. Mostly these days (actually the last 3 months or so), I build Allison V-1710 engines with a friend who is the best Allison guy in the world. Our engines are flying in P-38's, P-39's. P-40's. P-63's, Yak-3's, and are running in tractors and boats around the world. We recently finished the engine for Art Arfons' old "Green Monster" dragster (restored and currently owned by John Rolley of Tucson). We supplied the Allisons for Rod Lewis P-38F "Glacier Girl." Our Allisons are flying in the USA, England, Australia, New Zealand, and France as you read this.
Force MUST be applied at some point to be effective, and my contention, that worked on the Space Shuttle and various military missiles and rockets (including the Navy Standard Missile / Aegis) , is that the force MUST be applied through the engine mount, and both the pressure differential at the nozzle throat-exit and mass flow rate / velocity together supply the propulsive force of any reaction engine, acting through the engine case itself. If that is not the real case, then why is the engine bolted securely into the engine mount?
This is frustrating from a logical standpoint, but OK. Perhaps it IS a good time to retire and work on old WWII Allison engines for fun. It actually IS fun, and I don't have to deal with difficult students anymore. Let's just say the Engguy and I disagree, and probably will until his designs come to fruition and start working in the real world. At that time, we'll probably agree and have a pleasant beer together.
I'm waiting for his 12,000 HP aero engine. I have no doubt he can design and build a 12,000 HP engine ... but I don't think it will fly (actually get an airframe airborne).