# Relationship between momentum and energy for rocket-powered aircraft

I'm trying to write a problem for my students in an algebra-based physics class. We have a flight simulator and I've got a glider in the simulator that has a couple rocket boosters attached. Students know the mass of the glider, the thrust of the boosters, and the burn time of the boosters. The mass change of the boosters is negligible compared to the total mass of the glider-booster system. Students can fly the glider in the simulator, ignite the boosters, and watch the plane climb and accelerate. The simulator can export telemetry data (velocity and altitude) during the burn.

I want to ask the students how efficient the boosters are. We're learning about energy, and the students can compute the total change in energy between the start and end of the rocket burn $$\Delta E_{T}=\Delta E_{P} + \Delta E_{K}=mg(h_2 - h_1) + \frac{1}{2}m(v_2^2-v_1^2)$$ And if they knew how much work the boosters did on the glider, they could compute how much energy was lost to drag by subtracting the measured increase in energy from the total work done by the boosters. $$W_d = W_{boosters}-\Delta E_T$$ The problem comes in that the work equation is integral $$W = \int \textbf{F}\cdot d\textbf{s}$$ And the flight path of the glider, in this instance, is curved when velocity and altitude change simultaneously, so the only way to do this accurately is to integrate the pathlength.

If we could hold the altitude of the glider constant, so that it was just accelerating and not also climbing, the change in total energy simplifies to the change in kinetic energy. Students can compute how much momentum the boosters impart on the glider using the thrust and burn time $$\textbf{F}_T \Delta t=\Delta \textbf{p}=m \Delta \textbf{v}$$ And use the $\Delta$v to compute the change in kinetic energy that would be predicted in a frictionless vacuum. Then, computing the actual change in kinetic energy, they could determine the work done by drag. $$W_d = \frac{1}{2}m\big[\big((v_1+\Delta v)^2 - v_1^2 \big) - (v_2^2-v_1^2)\big]$$

Similarly, if we could hold the speed of the glider constant, and have it climb at that steady speed, the path of the glider would be a straight line. We could determine its length with the starting velocity and the burn time, use the work equation to determine the energy added to the system by the boosters and subtract the measured increase in energy due to altitude gain to determine the work done by drag. $$W_d =F_Tv_{1=2}\Delta t -mg\Delta h$$

But there's no easy way to ensure that the glider will maintain constant speed or constant velocity during the burn as there are no altitude-hold or speed-hold controllers built in. Students can't be expected to have the piloting skills to do this themselves either so I was kind of hoping they could do this exercise stick-free.

But is there a way to determine the work done by the boosters in the general case where both altitude and velocity are allowed to change without integrating? Am I missing something here? I haven't run the numbers but I suspect the amount of energy added by the boosters is different in the altitude-hold case vs the speed-hold case. Rockets don't add a fixed amount of energy to a system, they add a fixed amount of momentum.

• No offense, but you are using a particularly unsuitable system to teach these things. Just because you have a toy simulator for a technical system that has almost no uses other than as short to mid range weapons system, does not mean that it's a good way of teaching the basics about energy and momentum. May I suggest that you abandon this approach and use a more conventional way of teaching? It will greatly profit your students, if you do. – CuriousOne Dec 12 '14 at 23:51
• @CuriousOne That's a valid criticism. In actually, I work for the company that makes the flight simulator and I was asked to make some prototype examples of where the simulator could be used as a supplementary tool alongside a traditional curriculum (in math and physics) to engage students and make connections to "engineering outcomes" as they call them. So the larger design activity involves designing a vehicle and piloting it through an obstacle course but understanding potential and kinetic energy as well as rocket boosters is necessary for proper design and piloting decisions. – Sheldon Dec 16 '14 at 21:37
• sorry for your job requiring to come up with an application for a technology... that's always a bad place to be in. But seriously, that's simply not how good physics teaching works. I don't expect you to be able to convince your boss that this ain't going to fly... once money clouds the minds, all bets for a good outcome of this kind of activity are off. Having said that, I wish you personally that you can make the best of it! – CuriousOne Dec 17 '14 at 2:10