# File - math-ES-project-T1

```MATH (ES) PROJECT
Done by:
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-ofthe-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
• 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 š„ is a quantity
between 0 and 1 known as the traffic intensity. Find the rate of
change of the number of airplanes waiting with respect to the traffic
intensity when š„ = 0.1 and š„ = 0.7
Waiting at the airport
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 š” is measured 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
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?
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
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, šs is the frequency of the source, š£ is the velocity
of the waves, Všæ is the velocity of the listener and Vš  the velocity of
the source.
• Find an equation of the š¹ in term of Vš . Then find the rate of change
of š¹ with respect to Vš , In two cases once 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