Done by: 1-HAMAD ALI SALEH ALZAABI.2-AHMED ALI ALBADI.Grade:12 Section:07

MATH (ES) PROJECT

Objective

• This project covers the application of the early topics in calculus(Differentiation and its applications)

• This project covers:

TASK 1: Internet Search( Airbus A 380)

1- History

2- Airbus A380 characteristics.

3-How long it takes an A 380 to takeoff.

4-What is the takeoff velocity?

5-What is the takeoff distance,i.e. the runway length that will be crossed before it takes off?

1- History

• The Airbus A380 development programe was officially launched in June 1994, with development of the A380. and the Airbus A380 had a complex gestation which at one time involved a joint study with Boeing  on a 600-seat Very Large Commercial Transport.

• At one time, Airbus even considered a flying wing design for the A380, but this was abandoned for more traditional configurations.

• The Airbus A380 made its first transatlantic flight to Medellin, Colombia, on 10 January 2006 to test engine performance at high airport altitude.

2- Airbus A380 characteristics.

• The luxury Airbus A380 features stunning interiors with state-of-the-art entertainment and business facilities. • The A380, is reportedly the world's largest passenger aircraft that features 50 percent more floor space than its nearest competitor aircraft. Each twin-aisle cabin features wider seats, mood lighting for relaxation.

3-How long it takes an A 380 to takeoff. • It will take 78 seconds to take off.

4-What is the takeoff velocity? • Take off velocity = 280 km/hr in the direction of the runway

5-What is the takeoff distance.the runway length that will be crossed before it takes off?

2800 m

TASK 2

• Questions :

• 1-Waiting at the airport • 2-Taking-off Distance and Time • 3- Speed of an airplane by Radar • 4- Aircraft Glide Path • 5-Shock wave : Doppler Effect

Waiting at the airport • Question :

The average number of airplanes waiting in a modern airport to take off can be modeled by the function where is a quantity between 0 and 1 known as 𝑥the traffic intensity. Find the rate of change of the number of airplanes waiting with respect to the traffic intensity when = 0.1 and = 0.7𝑥 𝑥

Waiting at the airport

• Answer:

Taking-off Distance and Time • Question:

Suppose that the distance an Airbus travels along a runway before taking off is given by meter measured from the starting point, and is 𝑡measured in seconds from the time the brakes are released. The aircraft will become airborne when its speed reaches 72 meters per second. How long will it take to become airborne, and what distance will it travel in that time?

Taking-off Distance and Time

• Answer:

Speed of an airplane by Radar • Question: If this airplane was flying on a flight path that would take it directly over a radar station, as shown in figure below. If “s” was decreasing at a rate of 400 miles per hour when “s=10miles” what is the speed of the plane?

• Answer:

So the velocity of the plane is = 500 mi/h

Aircraft Glide Path Question:

Now the Airbus starts descent from an altitude of 1 mile, 4 miles west of the runway.

Find the cubic function fx=ax3+bx2+cx+d on the interval (-4,0) for the landing.

The function you founded models the glide path of the plane. When would the plane be descending at the greatest rate?

Aircraft Glide Path • Answer:

1 = a (-4)3 + b (-4)2 + c (-4) + (0) [ d = 0 ]

1 = -64 a + 16 b + -4 c

Starts descent at (-4, 1) so the slope before this point is =0

f(x) = ax3 + bx2 + cx + d

f ’(x) = 3ax2 + 2bx + c

0 = 3ax2 + 2bx + c

at x = 0, and x = -4, f ‘(x) = 0 , c = 0 • 0 = 3ax2 + 2bx + c• 0 = 3a(-4)2 + 2b(-4) + (0)• 0 = 48 a – 8 b

Next slide

Aircraft Glide Path (Answer) (Part A)

Aircraft Glide Path (Answer) (Part A)

Aircraft Glide Path (Answer) (Part B)

Shock wave : Doppler Effect• Question:

• If the frequency F of an airbus landing on runway heard by a stationary observer on F=v+vLv+vsfs where F is frequency perceived by the observer, 𝑓s is the frequency of the source, is the velocity of the waves, V is the 𝑣 𝐿velocity of the listener and V the velocity of the source. 𝑠• Find an equation of the in term of V . Then find the 𝐹 𝑠rate of change of with respect to V , In two cases 𝐹 𝑠once when airbus is approaching at velocity of 200km/hr, and the other when its moving away at a velocity 200km/hr

Shock wave : Doppler Effect• Answer:

Done by

• Grade :12 Section :07

• Names :

•1- HAMAD ALI SALEH ALZAABI•2- AHMED ALI ALBADI