A question on conservation of energy

[Born2Perform]

it is a basic question:

a spacetaxi must bring some people from planet A to planet B. wasting energy, it accelerates until the middle of the distance, then again using energy it decelerates to reach planet B. then it returns to planet A, in the same way.
it consumed fuel 4 times, but in space nothing is changed, and the momentum of the taxi returned 0.

how the energy is conserved here? thank you.

 

[physics.guru]

This is a brief explanation. If you need a more detailed mathematical solution, just let me know.

There are different types of energy (i.e. spring potential, kinetic, gravitational potential, potential chemical energy). Let's assume the fuel used by this space taxi is stored chemical energy). Also let's assume that during fuel consumption, no heat is lost and there is a direct tranformation of chemical energy into kinetic energy. Before the taxi begins its journey there is a certain amount of fuel available and thus a certain amount of chemical energy E_c available to use.

As the taxi makes its trip, some E_c is converted to kinetic energy K. We also assume the taxi uses a propulsion type system (such as in a rocket). This type of system relies on Conservation of Momentum. Think of a perfectly inelastic collision only in reverse. The chemical energy is used to accelerate the taxi according to the scenerio you have proposed. By the work energy theorem, the work done by the stored chemical energy over the entire trip can be summed by adding up the work done during each stage according to the work-energy theorem (W = change in K)
In short:

E_c = 4E_{chemical used} + remaining E_c

or

Delta E_c = 4E_{chemical used}

Where the 4 represents the four stages of the trip.

Keep in minde we've made several assumptions for simplification, but even without these assumptions, if we can measure everywhere energy is used and sum the results, then the latter equation holds true.

 

[Born2Perform]

hi: i didn't get the point.

$$E_c = 4E_c + remaining E_c$$

ok i got this, $$4E_c$$ is used to make kinetik energy of the ship to increase, but at the end of the trasformation, the kinetik energy of the ship returnes to 0, and the chemical energy is wasted the same.

with an analougue case, i can say that a second ship accelerates 4 times, instead to accelerate and decelerate 2 times like the example. so that second ship acquires a kinetic energy proportional to the $$4E_c$$ used, and the energy is spent good.
but if the total kinetic energy of the first ship at the end is 0, why energy is not 100% wasted?
thank you

 

[Doc Al]

While the KE of the ship ends up with no net change, what about the KE of the exhaust gases of the rocket thrusters that accelerated the ship? That's where the energy goes.