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.
