Q:

If the energy of an aircraft is the sum of its Kinetic Energ and Potential Energy, what does reducing the aircraft's weight do?

As per the title, if the energy of an aircraft is the sum of its Kinetic Energy,$=\frac12 mv^2$, and Potential Energy,$=mgh$, Does reducing its mass increase its energy?
I am trying to write an explanation why using a lighter material on aircraft is a good idea, and one reason is to reduce it's overall weight - but what effect of reducing it's weight does it have? I can't get it in to words which make sense. It will obviously use less fuel etc, but why - which formulas come in to play?
Thanks a lot.

A:

I am trying to write an explanation why using a lighter material on aircraft is a good idea, and one reason is to reduce it's overall weight - but what effect of reducing it's weight does it have?

In order to get an airplane from stationary on the tarmac to cruising speed $v$ and cruising altitude $h$ we have to spend energy:
$$T=\frac12 mv^2+mgh$$
This is even without taking into account the energy losses due to air drag.
To achieve this, thrust $\vec{F}$ is applied by the engines, causing acceleration $\vec{a}$:
$$\vec{F}=m\vec{a}$$
This force perfoms work $W$ on the plane ($\vec{r}$ is the position vector):
$$W=\int_{\mathrm{trajectory}}\vec{F}\mathrm{d}\vec{r}=\int_{\mathrm{trajectory}}m\vec{a}\mathrm{d}\vec{r}$$
Assuming $m$ is constant:
$$W=m\int_{\mathrm{trajectory}}\vec{a}\mathrm{d}\vec{r}$$
(Note again that work done against drag forces is not accounted for here. These condiderably increase the amount of work needed)
So this amount $W$ is proportional to the mass $m$ and is of course directly proportional to the amount of fuel used.
Once the plane has achieved cruising speed and altitude, only work done against the air drag forces are needed to maintain constant flight conditions:
$$W=\int_{\mathrm{trajectory}}\vec{F}_{drag}\mathrm{d}\vec{r}$$
$\vec{F}_{drag}$ is usually modelled as:
$$F_{drag}=\frac{1}{2} \rho C_D A v^2,$$
where $A$ is the cross section of the plane. So while $F_{drag}$ does not depend directly on mass, it does depend on plane size.

