# What Makes Airplanes Fly? History, Science and Applications ofAerodynamics

## CONTENTS OF CURRICULUM UNIT 90.07.10

- Narrative
- I. Introduction
- II. Rationale and General Objectives of the Unit
- III. Historical Overview of the Development of Aircraft4
- IV. The Mathematical Application
- II. An Introduction to Graph Theory
- Sample Lesson Plan 1
- Sample Lesson plan 2
- Sample Lesson Plan 3
- Sample Lesson Plan 4
- Sample Problems For Class Discussion
- Bibliography

### Unit Guide

## Historical Developments of the Aircraft Industry with Mathematical Applications

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## II. An Introduction to Graph Theory

Just as model aircraft are used to represent the real thing, models in this section will be used as a representation of something else. The model or graph will be used to give us an idea of what reality is.

A graph is a finite set of points, called vertices, together with a finite set of curved or straight connecting lines called edges, each of which joins a pair of vertices. These vertices and edges satisfy the condition that no edge begins or ends at the same vertex. Graphs without edges are called null graphs.

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The figure in c) is not a graph because it violates the condition that no end may join a vertex to itself. When a graph has two or more different edges joining the same pair of vertices, these edges are called multiple edges.

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Graphs are important tools in representing a vast number of real world problems. Graphs that are used to represent the lay out of streets are called street networks. In this instance the edges of the graph can have a direction indicated by arrow. These graphs are called digraphs. A digraph is a finite, non empty set of points called vertices, together with some directed edges joining these points. These edges are subjected to one restriction. The initial and terminal vertices of a directed edge may not be the same.

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### A. Graphs and Matrices: Matrices from Drawings

Matrices are presented as a means of storing information in which the position of the information is very important. The direct route matrix is used since it enables us to predict properties of more complicated networks without reproducing them.The figure shows a network of roads

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One way of describing a direct route linking endpoints is to use an arrow diagram.

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The diagram shows a network of roads between several towns with their direct routes.

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B. Finding the Shortest Path

A graph or network is defined by two sets of symbols, nodes and arcs. Nodes are the set of pints or vertices, arcs consists of an ordered pair of vertices and represents a possible direction of notion that may occur between vertices.

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DIJKSTRA’S Algorithm:10 This algorithm for finding the shortest path between a pair of nodes requires that all the arcs in the network have non-negative arc length.

The algorithm uses this method:

Let us use this method in the following example:

- 1. Designate node one as the starting point.
- 2. Find the node closest to node one.
- 3. Find the second closest node to node one.
- 4. Find the third closest node to node one.
- 5. Continue until all the paths are used.

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PATH | LENGTH OF PATH |

The arc (1,2) | 4 | |

The shortest path from node 1 | ||

to node 3 + arc (3,5) | 3 + 3 = 6 |

____ The shortest path from 1 to 3

+ arc (3,5) | 3 + 3 = 6 | |

Shortest path from 1 to 2 | ||

+ arc (2,5) | 4 + 2 = 6 | |

Shortest path from 1 to 2 | ||

+ arc (2,5) | 4 + 3 = 7 |

____ Shortest path from 1 to 2

+ arc (2,4) | 4 + 3 = 7 | |

Shortest path from 1 to 5 | ||

+ arc (5,6) | 6 + 2 = 8 |

____ Shortest path from 1 to 4

+ arc(5,6) | 7 + 2 = 9 | |

Shortest path from 1 to 5 | ||

+ arc(5,6) | 6 + 2 = 8 |

Summary of shortest path

Nodes | Closest Nodes | Path from Node 1 to | Length of | |

to node 1 | the nth closet node | path | ||

0 | 1 | - | - | |

1 | 3 | 1 - 3 | 3 | |

2 | 2 | 1 - 2 | 4 | |

3 | 5 | 1 - 2 - 5 or 1-3-5 | 6 | |

4 | 4 | 1 - 2 - 4 | 7 | |

5 | 6 | 1-3-5-6 or 1-2-5-6 | 8 |

The digraph can also be used to solve complicated problems. Consider this problem: We wish to minimize the time an aircraft spends at an airport. The component activities can be placed in a table:

A1 | Disembark passengers | 1/2 hr. |

A2 | Unload baggage | 1 hr |

A3 | Clean the plane | 1/2 hr |

A4 | Take on new passengers | 1 hr. |

A5 | Load new baggage | 1 hr. |

An Activity Analysis Digraph is constructed in the following way.

Now that we have drawn a model, the problem is to determine the shortest time for the completion of the whole job. We can proceed as follows:

- 1. Represent each activity by a node A1, A2, . . . An with the time required for the activity.
- 2. Create two additional nodes each labeled with the number zero. One representing the job’s beginning and the other the job’s end.
- 3. Draw a directed edge from one activity to the next only if the first activity precedes the second.

The critical path is the path of longest time from B to E. To determine the most efficient schedule in the problem is the critical path B, A2, A4, which has length of two hours and this gives the minimum time for the whole job to be completed.

- 1. Denote time t measured from starting point B; t = 0
- 2. Rephrase the problem; given an Activity Analysis Digraph for a project. What will be the shortest time at which E, the end, can be completed?
- 3. Add the times for all activities on the path up to but not including E (There may be more than one path from B to E).

To explain the activities A1 and A2 can both be started at time zero (Passengers can disembark and luggage be taken off at the same time).

The activity A3 cannot be started until all the passengers are taken off; the activity A4 cannot be started until A3 is completed but can be made to overlap with A5; we cannot arrive at E until both A4 and A5 are completed.