Here is a theoretical example where the energy into the system does not mach the energy output.
There is no friction or air resistance. A sledge with a rocket engine attached to it is lying still on the ground, some distance in front of a ramp. The ramp has a shape such that, if the sledge drives onto the ramp, it will be thrown vertically upwards. The sledge and the engine have a mass of 1000kg, the engine has a power output of 10kN. The fuel usage is 1L/s. The fuel's mass is too low to be included in the total mass.
The sledge accelerates towards the ramp, until it runs out of fuel. It then hits the ramp, and is thrown upwards. The potential energy the sledge will have at the highest point after the jump, will be proportional to the height the sledge will reach (E = mgh). The potential energy at the highest point will be proportional to the kinetic energy the sledge had before it ran out of fuel, which will be proportional to the energy in the fuel in the engine at the start. Therefore the height the sledge will reach will be proportional to the fuel in the engine at the start.
I do two theoretical tests. First I start with 1L of fuel, the with 2L of fuel. I expect that since I double the fuel, I will also double the maximum height.
Test 1. I let the fuel be 1L. fuel_1 = 1L fuelPerSec = 1L/s g = 9.81 m/s^2 ≈ 10m/s^2 m = 1000kg f = 10kN v_0 = 0m/s a = f/m = 10kN / 1000kg = 10m/s^2 t_1 = fuel_1 / fuelPerSec = 1L / ( 1L/s ) = 1s v_1 = a*t_1 = 10m/s2 * 1s = 10m/s Ek = 1/2*m*v_1^2 = 1/2 * 1000kg * (10m/s)2 = 50kJ Ep = Ek = 50kJ Ep = mgh i.e h = Ep / (mg) = 50kJ / (10m/s^2 * 1000kg) = 5m Test 2. I let the fuel be 2L. fuel_2 = 2L fuelPerSec = 1L/s g = 9.81 m/s^2 ≈ 10m/s^2 m = 1000kg f = 10kN v_0 = 0m/s a = f/m = 10kN / 1000kg = 10m/s^2 t_1 = fuel_1 / fuelPerSec = 2L / ( 1L/s ) = 2s v_1 = a*t_1 = 10m/s2 * 2s = 20m/s Ek = 1/2*m*v_1^2 = 1/2 * 1000kg * (20m/s)2 = 200kJ Ep = Ek = 200kJ Ep = mgh i.e h = Ep / (mg) = 200kJ / (10m/s^2 * 1000kg) = 20m
When I double the energy input, the energy output is quadrupled. The relation should be proportional but it is quadratic.