I love these discussions. My first CO always liked to pose questions like these during All Officer Meetings and it's surpising the wide variety of interpretations and misinterpretations. One of his favorites was to ask, "assuming an F14 can fly at the speed of light and it turns its landing light on...what happens to the light? A...the light goes forward at twice the speed of light. B...the light doesn't shine at all. or C...the light goes backwards like a shock wave. I, being practical kinda guy, chose D....you rip your landing gear off.
Anyway....in these discussions you have to realize that in the real world you cannot isolate one single factor, such as momentum, and achieve much insight for the pilot into the total picture of what an airplane will do. That's why concepts and formulas such as Ps become important because they take the total of the equation including mass, thrust, drag, gravity, etc., into account to tell you what all the individual factors taken together really mean to a guy in the cockpit.
Also, a zoom climb without power is meaningless unless you're low, your engine quits and you need to bailout/eject. Overall though, other individual forces are combined in the significance of Ps. That goes to my original comment regarding Humble's suggestion regarding E=MC2. In isolation, total E doesn't really mean much, it's only when it's put to use within the context of the rest of the aerodynamic and physics equations that much meaningful information is gained. Even the idea of Energy Height is only useful as an academic means to compare two aircraft (in a relatively inaccurate comparison at that) or to use as basic instruction regarding the transformation between potential and kinetic energy but is otherwise meaningless to the guy in the cockpit because it assumes no losses and no atmosphere.
As far as the discussion regarding a dive then climb, some are missing other, more important, details while focusing on momentum. Let's take the extremes for illustration to prove the point. We start again with two identical aircraft but of different mass. For simplicity we'll assume zero thrust and drag. Both dive and then pull 4g across the bottom into a zoom. The conclusion that some have reached is that the additional weight aides the downward acceleration and this "stored" momentum helps the heavier aircraft outzoom the other.
First, the idea of "extra" acceleration in the dive is just wrong but we'll define the question as "in a pure vertical maneuver (dive or zoom), which airplane outperforms the other?" Here's a very basic point of view we should all be familiar with even without having taken physics and calculus. Acceleration in the dive and deceleration in the zoom due to gravity is identical for both, and it is completely independant on their relative weights/mass. Both planes will reach the same speeds and distance traveled in either a dive or zoom at the same time. For reference see: "Pisa, Leaning Tower Of" and "Galileo Galilli". Specifically regarding momentum (P=mV), we hold mass constant (I figure that at these speeds the conversion of mass to energy per Einstein is rather small) and the only variable is velocity and, since Galileo proved the velocity for both airplanes will be the same at each point in the curve, then momentum of the dive and climb cancel out.
More relevant to the real world and as others have pointed out, there's that "identical" 4g pull which isn't really the same for both aircraft weights. The heavier one has to pull a higher AOA to achieve the same 4G pull and creates much greater drag and E loss in the pull.
So, the dive/climb question comes out to this. Assuming identical aircraft except for weight the heavier aircraft will not zoom as high as the lighter aircraft because of losses due to drag during the pull and higher drag during any portion in which lift is being generated. That leaves us with a negative energy aspect (drag) which can only be offset by a positive energy aspect (thrust) and the heavier aircraft has inferior Ps.
For those trying to test this, the problem is most likely in technique vice physics. Pilot technique is essential when coming up with repeatable data. Small factors such as an acceleration or deceleration prior to a dive. The consistency and smoothness of a pull. Trim, finding pure vertical, etc., all makes flight test the challenge that it is.
