# Applied Mathematics: aircraft landing process

Updated on April 3, 2016

## Use Of Theory In Engineering

Engineers use scientific and mathematical theory as the foundation of all designs. Scientific approach ensures repeatability of process.

Newton Laws Of Motion

1. A body remains in its state of motion or rest until acted upon a resultant force.
2. The rate of change of momentum is equal to the resultant force and acts in the direction of force.
3. For every action there is an equal and opposite reaction

## Landing Aircraft Runway Required

Civil Engineers use the basic Newtonian mechanics to analyze and calculate the runway length for landing aircraft.

Moreover the process is used to determine the locations of exit taxiways on a given runway. An aircraft is treated as classical particle, that of a point mass. This method is elegant and widely applied and is adopted by the International Civil Aviation Organization, (ICAO), the UN technical organ for aviation.

Newton's Laws of motion are applied.

## Safe and Optimal Use Of Runways

An aircraft touches down on the runway in the touchdown zone after it achieves the so called landing configuration. After touchdown it stabilizes and starts the deceleration process to reduce speed.

The aircraft then exists the runway into the taxiway after reaching a safe and comfortable speed. Only after leaving the runway strip will the runway be freed for a subsequent landing or a take-off.

Airport authorities are concerned with twin factors safe utilization of the runway and optimum runway utilization factor.

Civil Engineers design the airfield in a way to ensure that the aircraft exists the runway as quickly as is safely and comfortably to do so.

## Kinematics Of Aircraft Landing

The kinematic equation:

v2 = u2 + 2as ,

is used to determine the exit locations on the runway for an aircraft.

• v is the final velocity, this is the safe and comfortable speed at which an aircraft is permitted to exit the runway
• u is the initial velocity, this is the touch down speed or the speed at which an aircraft has stabilized after touchdown to start braking
• a is the acceleration (braking)
• s is the distance travelled from touchdown point to the point at which it is permitted to exult the runway (the unknown).

This is a typical Newtonian equation of motion with terms used in their usual form. Moreover each aircraft has the terms v, u, and a specified by the manufacturer in the technical manuals.

This equation is solved in the usual was of substitution and making s the subject of the formula.

For an airport serving several different type of aircraft spreadsheets are used to solve the multiple equations and determine various locations where aircraft in the same category can be allied to exit the runway.

## Tamplate For Aircraft Exist Taxiway Requirement

Aircraft
Performance (v,u,a) From Manual
s = (v2 - u2)/2
B747

A380

Dash8

A321

B737

## Use Of Spreadsheets In Engineering

The application of computers in engineering is exemplified by this method. The table above is modeled in a spreadsheet such as MS Excel to calculate the safe exit locations of a fleet that uses an airport.

The aircraft are grouped according to their relative safe runway exist locations and optimal exit taxiways are determined for a runway under design.

19

4

4

13

31

3

24