I'm working on a simple rocket flight simulation program, and am trying to reproduce aspects of Goddard's first successful rocket flight. My sources put the empty weight at 5.75-6lbs, loaded weight 10.25-10.4 lbs, and thrust 9 lbs (40N), using a fuel-rich gasoline/LOX propellant with Isp of about 150s. The rocket is supposed to have reached an altitude of 41' (12.5m) and turned to cover 184' (56m) horizontally in a 2.5 sec flight after burning 20 seconds without leaving the launch stand. Fully fueled, the rocket didn't reach 1:1 thrust-to-weight, so couldn't lift off; my calculations agree that 20-25 seconds of fuel consumption is about where the weight would have dropped to 9 lbs.; so far so good.
However, in order to reach 12.5m altitude in no more than 2.5 seconds, the rocket would have to accelerate at about 4m/s² on average (d = 1/2 at²), implying thrust-to-weight ratio of around 1.4; this thrust could be achieved only very briefly right before fuel-out, and that would leave no time at all for the fall (actually a powered dive!) from that peak altitude.
If we take the widely reported 41' altitude as counting the rocket's 10' height, that still leaves us with a 9.5m apogee to account for.
My simulation doesn't yet include rotation off the vertical, so my final apogee is much higher (Goddard's rocket had another 40 or 50 seconds of propellant available after liftoff; it flipped over and crashed quite quickly), but 2.5 seconds after liftoff I see only 0.04m of vertical travel, nowhere near the reported figure.
Am I overlooking something obvious? My model includes drag, but at the low speeds in the first seconds of travel, it's negligible.
My best guess at this point is that the nozzle burnthrough that caused the rocket to go off-course also greatly increased the thrust while decreasing the Isp (i.e. it let a lot of unburned propellant out of the combustion chamber in a hurry); if I change thrust to 100N and Isp to 50s right at takeoff, my simulation can reach the reported altitude at the required time. Does that seem plausible?