How can the simple force balance be refined to account for the change in momentum during flight? How significant is this difference?

The idea behind this project is to investigate the costs (economic, environmental,
or other) of aviation.
For example: what does it cost to fly a Boeing 747 from London to New York?
One suggested approach could be first to consider just the cruising phase of the
journey, and that only four main forces are involved: lift, drag, thrust and weight.
Assuming that all of the forces balance, a simple ordinary differential equation
that describes fuel usage can be obtained, which can be solved analytically to give
an expression for the fuel required to fly a given distance.
The assumption of balancing horizontal forces is not strictly correct; a refined
approach would use an appropriate application of Newtons second law to correctly
account for the change in momentum due to burning fuel.
The important part of the study will be to explain the model and consider the
consequences of the result.
Possible questions that could be addressed include (but not limited to):
How much fuel does it take to fly a Boeing 747 from London to New York?
How can the simple force balance be refined to account for the change in momentum
during flight? How significant is this difference? What is the difference in
fuel requirements flying from London to New York or Edinburgh? Why do aeroplanes
fly at altitude? What are the consequences of flying higher/lower? Is there
any difference making the return trip without refuelling? What difference does
wind make? What are the consequences for engine design? Are baggage charges
justified? How do the different aspects of a journey contribute to the overall cost?
(e.g. how do take-off, climbing, and cruising etc. compare with the total cost?) Is
the airlines choice (size) of aircraft related to the distance travelled? What affects
the length of a runway? How do the various costs compare with other forms of
transport? How does the content of the plane matter? What kind of balance is
there between economic and environmental (or any other) costs? Would a 747 be
able to fly on Mars?
References. Nakayama and Boucher Introduction to Fluid Mechanics