Consider an aircraft at point A, moving along a curved flight path. From Newton's 2nd Law of Motion, summing forces parallel and perpendicular to the flight path,
parallel to
the flight path.
perpendicular
to the flight path
and
Here we still stick to the motion of an airplane at relatively slow speeds (not a high supersonic Mach number) so that we can neglect the effects of the curvature of Earth.
Under Performance, we think about issues like:
If acceleration =0, we get Static Performance: range, endurance, maximum speed etc.,
needed for aircraft design and operations.
Let 
and
acceleration =0.
for most aircraft,
so T=D; L=W for level, unaccelerated flight.
Thrust required for steady level flight
;
.
So, 
, or 
.
Hence,
. If 
; then 
, or 
.
. Thus 
.
Maximum Rate of Climb:
. Depends on altitude.
T=0 : Equilibrium: no acceleration.
(parallel)
(perpendicular).
Hence,
where 
is the glide path angle.
Rate of sink = 
=
.
Minimum Rate of Sink 
occurs at 
Takeoff Distance:
Net Thrust is the thrust minus the ground roll friction, drag etc.
Let's assume, for example, that Net Thrust is 0.2 Wto, where Wto is the takeoff
weight of the aircraft. Then, 
.
Kinetic Energy = 
where R is the takeoff run.
This says that in distance R, we gained enough kinetic energy to be at the takeoff speed, accelerating at the rate corresponding to net thrust of 0.2Wto.
Thus, 
.
Runway length should be twice this distance, in order to provide enough distance to stop if the decision to abort takeoff is made at the takeoff speed.
.Now 
.
So, distance traveled per unit mass of fuel consumed = 
.
Range =
.
Note: this is strictly valid only for propeller-driven airplanes. Here we will use it as a first approximation. More detailed treatments will be left to the 3rd-year Flight Vehicle Performance course
We will assume that the fuel consumption in descent and landing is at the same rate as during cruise (this is conservative).
For the climb phase, we will assume the fuel consumption for cruise plus an increment depending on the cruise altitude. The table below is from Shevell, and is constructed from data on large commercial airliners.
Typical landing procedure for an airliner:
Descent to 5000 feet. Vectored to 12 miles downwind, make a 180-degree turn. Extend flaps and landing gear, reduce speed to 150mph.
This leaves 5 minutes of final approach to do the flap deflections, landing gear deployment, lining up with the runway etc..
The actual landing process (note: based on experience with a small plane plus a lurid imagination about what is must be like in larger planes) consists of something like the following:
1. Maintain constant angle of descent along a specified flight path (staying along the middle of the "glide slope"). Align with the runway center line.
2. Arriving over the runway end, reduce power, and increase angle of attack to level off with the landing gear a few feet off the ground. Use ground effect to continue flying essentially level, as the speed reduces, and the aircraft slowly sinks to the ground.
3. Just before touchdown, flare (increase angle of attack, nearly to stall), and hold attitude. The rate of sink comes to very near zero, and as the speed comes below the stalling speed in ground effect, the main-landing-gear wheels touch the ground. Continue flying and bring nose slowly down until nose gear touches.
4. Use reverse thrust if any to slow down, otherwise roll with light braking, slow down to the speed for turning off the runway, and take the next taxiway turnoff as instructed by the Tower (don't take any sports-car type corners, or you may run off into the grass, take out some landing lights, or worse :) ).
The above sounds routine, but factors like cross-winds make life much more interesting. In cross-winds, the airplane may be piloted down with the "controls crossed" (roll so that the lift vector is pointed into the wind, but use rudder in the opposite direction to keep from changing flight direction into the wind. As the airplane comes close to the runway, the wind will generally change quickly in speed and direction - and the pilot must compensate to keep the aircraft aligned with the runway. To land in cross-wind, the airplane continues to be held in a banked attitude, so that the landing gear on the side facing into the wind, touches down first, then the other side, and the pilot holds the aircraft from swerving on the runway due to the greatly increased drag on one side.
Add gusts to the cross-wind, and the need for piloting skills becomes much more evident.
Sometimes, to increase the rate of descent down to the runway, the pilot may fly with the aircraft actually sideslipping, alternately one way and then the other, and again controls crossed to induce a side-slipping motion. The reasoning here is that the drag is much more in sideslip, so the forward speed is reduced and the angle of descent made a lot steeper.
The
aircraft is considered to be a rigid body, with the lift (L), drag (D), thrust
(T) and weight (W) acting on it. Flight Dynamics deals with the movement of
the aircraft as it responds to these forces. Table: Empirical Guidance on Fuel Consumed During Climb from Takeoff at Sea-Level to a Given Cruising Altitude
|
Altitude
|
% of takeoff weight as added fuel consumption
|
|
20,000 feet
|
0.75%
|
|
30,000 feet
|
1.25%
|
|
35,000 feet
|
1.60%
|
|
|
Solids |
|
|
|
Flight Mechanics |
|
Manufacturing |
Table 1: Direct Access to the Sub-Disciplines of Aerospace Engineering
|
|
Solids |
|
|
|
Flight Mechanics |
|
Manufacturing |
