Take Off!!!Take Off!!!

Rosny DanielRosny Daniel

Daniel PappalardoDaniel Pappalardo

Nadeline RabotNadeline Rabot

IntroductionIntroduction

In this project we will discuss the In this project we will discuss the advantages of arresting cables and advantages of arresting cables and steam catapults on aircraft carriers steam catapults on aircraft carriers by comparing the distance needed by comparing the distance needed with and without these factors for with and without these factors for

take off and landing.take off and landing.

The Four Aerodynamic Forces The Four Aerodynamic Forces That Act Upon an Airplane in That Act Upon an Airplane in

FlightFlight

Weight

Thrust

Lift

Drag

F-15 Eagle: SpecificationsF-15 Eagle: Specifications

Weight: 40,000 lbs loadedWeight: 40,000 lbs loadedPowerplant: EnginePowerplant: Engine

Pratt and Whitney F100-229 Pratt and Whitney F100-229 Afterburning Turbo Fans Afterburning Turbo Fans 29,000 lbf29,000 lbf

Wing Area: 608 ftWing Area: 608 ft22

Max speed: 1,875 MPHMax speed: 1,875 MPHCruising Speed: 570 MPHCruising Speed: 570 MPH

Armament: Armament: Guns: 1 x M61 Vulcan 20 mm Guns: 1 x M61 Vulcan 20 mm

Gatling Gun w/ 940 Gatling Gun w/ 940 roundsroundsMissles: Combination of AIM-7F Missles: Combination of AIM-7F

Sparrows, AIM-120 Sparrows, AIM-120 AMRAAMS and AIM-9 AMRAAMS and AIM-9 SidewindersSidewinders

Take Off…Take Off…

Steam CatapultSteam Catapult

• Device used to launch the aircraft from Device used to launch the aircraft from aircraft carriersaircraft carriers

• Generally a track built into the flight deckGenerally a track built into the flight deck

• Below is a piston that attaches up through Below is a piston that attaches up through the track to the nose of the aircraftthe track to the nose of the aircraft

• At launch, a release bar holds the aircraft At launch, a release bar holds the aircraft in place as the steam pressure builds up in place as the steam pressure builds up and then it releases the aircraft at a high and then it releases the aircraft at a high speedspeed

Steam Catapult (continued)Steam Catapult (continued)

A catapult can accelerate an aircraft from 0 A catapult can accelerate an aircraft from 0 to 182 mph in 2 seconds. This allows an to 182 mph in 2 seconds. This allows an

aircraft to take off safely on a 300ft deck.aircraft to take off safely on a 300ft deck.

Stall VelocityStall VelocityStall Velocity is the minimal velocity necessary for the Stall Velocity is the minimal velocity necessary for the

aircraft to take flight or remain airborneaircraft to take flight or remain airborne

To solve for the stall velocity we need the Lift equation:To solve for the stall velocity we need the Lift equation:

L = (1/2)p * VL = (1/2)p * V2 2 * S* SRefRef * C * CLL

L = Lift forceL = Lift forceP = air density (.00237 slug/ ft)P = air density (.00237 slug/ ft)V = aircraft velocityV = aircraft velocity

SSRefRef = reference area (surface area of wings) = reference area (surface area of wings)

CCLL = Coefficient of lift (1.6) = Coefficient of lift (1.6)

Stall VelocityStall Velocity(continued)(continued)

We need enough lift to counteract We need enough lift to counteract the takeoff weight. So to solve for the takeoff weight. So to solve for the stall velocity we get:the stall velocity we get:

VVSS = (2*W/(p* S = (2*W/(p* SRefRef * C * CL L ))))1/21/2

Weight = 40,000 lbsWeight = 40,000 lbsP = .00237 slugs/ ftP = .00237 slugs/ ft33

SSRefRef = 605 ft = 605 ft22

CCLL = 1.6 for supersonic jets = 1.6 for supersonic jets

VVs s = ((2*44,000)/(.00237*605*1.6)) = ((2*44,000)/(.00237*605*1.6))1/21/2

VVs s = 186.7 MPH = 186.7 MPH

Thrust, Acceleration, Thrust, Acceleration, DistanceDistance

Thrust: Thrust: Pratt and Whitney F100-229 Afterburning Turbo Fan Engine Pratt and Whitney F100-229 Afterburning Turbo Fan Engine = 29,000 lbs of Thrust/ engine= 29,000 lbs of Thrust/ engine

1 pound of force = 4.44822162 Newtons1 pound of force = 4.44822162 Newtons2 Engines = 257,996.85 Newtons2 Engines = 257,996.85 Newtons

Thrust is a force, therefore Newton’s Second Law applies.Thrust is a force, therefore Newton’s Second Law applies.F = MAF = MA

257,996.85 = mass of F15 * acceleration257,996.85 = mass of F15 * acceleration257,996.85 = 18,181.8 kg * acceleration257,996.85 = 18,181.8 kg * accelerationacceleration = 14.2 m/sacceleration = 14.2 m/s22

acceleration = 46.576 ft/sacceleration = 46.576 ft/s22

Thrust, Acceleration, Distance Thrust, Acceleration, Distance (continued)(continued)

If we anti-differentiate the acceleration we will end up with the If we anti-differentiate the acceleration we will end up with the velocity equation. From here, we can solve for the amount velocity equation. From here, we can solve for the amount of time it will take the plane to reach the stall velocity and of time it will take the plane to reach the stall velocity and then the amount of distance it will take the aircraft to take then the amount of distance it will take the aircraft to take off.off.

A = 46.576 ft/sA = 46.576 ft/s22

V = 46.576 ft/s * time + VV = 46.576 ft/s * time + V00

186.7 MPH = 273.83 ft/s186.7 MPH = 273.83 ft/sVV00 = 0 = 0

273.83 = 46.576 * t273.83 = 46.576 * tt = 5.9 secondst = 5.9 seconds

Thrust, Acceleration, Distance Thrust, Acceleration, Distance (continued)(continued)

Velocity Vs. Time of an F15 During Take Off

y = 46.412x

0

50

100

150

200

250

300

0 1 2 3 4 5 6 7

Time (seconds)

Vel

oci

ty (ft

/s)

Velocity vs. Time

Linear (Velocity vs. Time)

By taking the integral of this function we find total displacement of the aircraft during takeoff. Calculating the area underneath the graph or Euler’s Method is another method of calculating the total displacement.

0∫5.9 46.412x dx = 807.8 ft

Landing…Landing…

Coming to a Coming to a STOPSTOP!!

Arresting CablesArresting Cables

ArrestingCable

Arresting CablesArresting Cables

• Thick steal cablesThick steal cables• Fitted at the end of the flight deck on an Fitted at the end of the flight deck on an

aircraft carrieraircraft carrier• Planes have a tail hook that catches onto Planes have a tail hook that catches onto

the cablethe cable• Cable takes up the slack by a hydraulic Cable takes up the slack by a hydraulic

mechanism that rapidly decelerates the mechanism that rapidly decelerates the aircraftaircraft

• Arresting cables stop F-15s within two Arresting cables stop F-15s within two seconds of engaging the cable, and within seconds of engaging the cable, and within 320 ft of touch down.320 ft of touch down.

Kinetic FrictionKinetic Friction

When the aircraft touches down, there are THREE main forces acting upon it: Normal Force from When the aircraft touches down, there are THREE main forces acting upon it: Normal Force from the runway, the Force of Gravity, and the Kinetic Friction Force.the runway, the Force of Gravity, and the Kinetic Friction Force.

The Normal force and the Force from gravity act upon the plane in the vertical axis. The net The Normal force and the Force from gravity act upon the plane in the vertical axis. The net force in the vertical direction equals zero because there is no acceleration of the plane in force in the vertical direction equals zero because there is no acceleration of the plane in the vertical direction. The only force acting upon the plane in the horizontal direction is the the vertical direction. The only force acting upon the plane in the horizontal direction is the force due to kinetic friction.force due to kinetic friction.

Newton’s 2Newton’s 2ndnd Law: Law:F = MAF = MAFFgg + F + FNN + F + FFF = MA; F = MA; Fgg = mg, g = 9.8 m/s = mg, g = 9.8 m/s

FFFF = - = -μμ F Fn ,n ,, , μμ = .5 = .5

Horizontal axis : -Horizontal axis : -μμ F FNN = Ma = Maxx

Vertical axis: -mg + FVertical axis: -mg + FNN = 0 = 0FFNN = Mg = Mg

--μμ Mg = Ma Mg = Maxx

--μμg = ag = axx

-.5*9.8 = -4.9 m/s-.5*9.8 = -4.9 m/s2 2 = a= axx

Landing the F-15Landing the F-15Velocity vs. Time

0

10

20

30

40

50

60

70

80

90

0 2 4 6 8 10 12 14 16 18 20

time (seconds)

velo

city

(m

eter

s/se

con

d)

Series1

The area under this curve is the total displacement of the F-15 after touchdown, thus the necessary length of the runway is at least 737.375 meters, or 2418.6 feet.

Riemann Sum for Velocity Riemann Sum for Velocity FunctionFunctionNumber Number of stepsof steps

Starting Starting pointpoint

Ending Ending pointpoint

Total area Total area under under curvecurve

11 00 17.3517.35 00

1010 00 17.3517.35 663.49663.49

5050 0 0 17.3517.35 722.5722.5

100100 00 17.3517.35 729.87729.87

250250 00 17.3517.35 734.29734.29

∞∞ 00 17.3517.35 737.375737.375

ConclusionsConclusions

To take off with the steam catapult, an F-To take off with the steam catapult, an F-15 fighter jet only needs 300ft of 15 fighter jet only needs 300ft of runway. We found that without the runway. We found that without the catapult, it will take at least 808ft to catapult, it will take at least 808ft to take off.take off.

To land with arresting cables, an F-15 only To land with arresting cables, an F-15 only needs 320ft of runway. We found that needs 320ft of runway. We found that without the arresting cables, it will take without the arresting cables, it will take at least 2419ft to land safely.at least 2419ft to land safely.

Conclusions (continued)Conclusions (continued)

Through applying and Through applying and using our extensive using our extensive knowledge of calculus, knowledge of calculus, and what it is related to, and what it is related to, we have decided that we have decided that the engineers in the the engineers in the United States United States Government and at Government and at N.A.S.A. are pretty N.A.S.A. are pretty smart and efficient. The smart and efficient. The inventions of the steam inventions of the steam catapult and arresting catapult and arresting cables are great. cables are great.

The End