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Performance Calculations

Introduction:

Units of Measure:

  • Statute Mile: the same as a standard mile as you would see driving a car
  • Nautical Mile: defined as one minute of arc along a meridian of the Earth. Using the widely accepted WGS84 ellipsoid model, this averages a nautical mile to 6,076 feet (1,852 meters), or 1.15 statute miles

Temperature Conversion Chart
Figure 1: Temperature Conversion Chart

Temperature Conversion:

  • The U.S. is used to operating on the Fahrenheit scale for day to day life but aviation standard is Celsius
    • Formula:
      • °C = [(°F - 32) x 5/9]
      • °F = [(°C x 9/5) + 32]
    • Example:
      • 70°F day
    • Calculate:
      • °C = ((70°F-32) x 5/9)
      • You should come out with 21.1°C
    • Chart [Figure 1]
      • Start at your initial temperature on the Fahrenheit scale
      • Move across until you hit the reference line
      • Move down and read the temperature off of the bottom
      • In this example it comes out to be roughly 22°C
    • Chart: [Figure 2]
      • Find the temperature you need and read across the appropriate column
      • Notice this chart is more designed for Celsius to Fahrenheit but we still come out just over 21°C

Temperature Conversion Chart
Figure 2: Temperature Conversion Chart


Crosswind Component:



Short Field Takeoff:

  • Use the chart for all performance data specific to an aircraft, in this example, a Cessna 172
  • Typically, there will be more than one chart for the same thing, separated by weight or aircraft configuration conditions
  • Always round up if your weight is not close to the reference weights they provide, this is because takeoff data will never improve with weight and therefore your numbers will be more conservative and provide a safety margin
    • Example:
      • Aircraft Weight: 2300lbs
      • Altitude: 3,000' MSL
      • 20° C
    • Chart: [Figure 4]
      • Starting at the left with the altitude, continue right across the chart until you reach the appropriate temperature
      • We expect a 1,100' takeoff without obstacles and 1,970' with a 50' obstacle
        • With a headwind of 9 knots, we can expect 990' takeoff without obstacles and 1,773' with a 50' obstacle
        • With a tailwind of 4 knots, we can expect 1,320' takeoff without obstacles and 2,364' with a 50' obstacle

Crosswind Component Chart
Figure 4: Short Field Takeoff Performance


Load Factor:

  • Load factor is generally not calculated as part of preflight however, it has a close relation to stall speed, which is very important
  • As load factor increase, stall speed increases
    • Formula:
      • Load Factor = 1 / cos(angle of bank)
    • Example:
      • Angle of Bank: 60°
    • Calculate:
      • Load Factor = 1 / cos(60)
      • Load Factor = 1 / 0.5
      • Load Factor = 2
    • Chart: [Figure 5]
      • Look at the 60° mark at the bottom of the chart and move up until you intercept the reference line
      • Move over to the left and see the load factor imposed on the aircraft
        • You should come up to approximately 2

Crosswind Component Chart
Figure 5: Load Factor vs. Stall Speed


Stall Speed Banked

  • Stall speed increases in a turn due to a loss in the vertical component of lift
    • Formula:
      • Stall Speed Banked = [Stall Speed Level / Cos (Bank Angle)]
    • Example:
      • Stall Speed: 48 KIAS
      • Bank Angle: 60°
    • Calculate:
      • Stall Speed Banked = [48 / Cos(60)]
      • Stall Speed Banked = [48 / 0.5]
      • Stall Speed Banked = 96 KIAS


True Vs. Magnetic North Course Conversion

  • Used primarily for flight planning when converting a chart (always true north) to a course to fly in the aircraft (magnetic north)
    • Formula:
      • "East is least, west is best"
      • Magnetic Course (MC) = True Course (TC) - East Variation
      • Magnetic Course (MC) = True Course (TC) + West Variation
    • Example:
      • True course is 270°
      • Variation is 14° east
    • Calculate:
      • MC = 270° - 14°
      • MC = 256°


Mach Number

  • Most high-speed aircraft are limited to a maximum Mach number at which they can fly
  • This is shown on a Machmeter as a decimal fraction
    • Mach 1 vs. Altitude
      Figure 6: AerospaceWeb.org. Mach 1 vs. Altitude

    • Formula:
      • Mach Number = Aircraft Speed/Speed of Sound (dependent on altitude)
    • Example:
      • Aircraft is flying at 30,000'
      • Speed of sound at 30,000' = 589.4 knots
      • The airspeed is 489.3 knots
    • Calculate:
      • 489.3/589.5 = 0.83 Mach


Pressure Altitude

  • As altitude increases pressure will decrease in a standard atmosphere
    • Formula:
      • Pressure Altitude = [(29.92 - current baro) * 1000] + Current field elevation
    • Example:
      • Current baro: 29.82
      • Field elevation: 500'
      Density Altitude Conversion Chart
      Figure 7: Pressure/Density Altitude Conversion Chart
    • Calculate:
      • 29.92-29.82 = .10
      • 0.10 * 1000 = 100'
      • 100' + 500' = 600'
    • Chart: [Figure 7]
      • Using the chart on the right of the graph, look for the current altimeter setting
      • To the right of it there will be an altitude in feet, and that is your conversion


Density Altitude

  • Pressure altitude corrected for non-standard temperature
  • Used for performance calculations
    • Formula:
      • Pressure Altitude + (120 x [Outside Air Temperature (OAT) - 15°C (ISA Temp)])
    • Example:
      • Pressure Altitude = 600' (as calculated above)
      • OAT: 10°C
    • Calculate:
      • 600' + [120 * (10-14)]
      • 600' + (-480) = 120'
    • Chart: [Figure 7]
      • From the temperature on the bottom move up to your pressure altitude
      • Next move left and read your density altitude off the scale
  • Other tools are available to help you calculate density altitude such as NOAA's Density Altitude Calculator


Cloud Bases

  • Used for VFR planning or when icing is a concern
  • This is a very rough formula as cloud bases are not always flat and can change rapidly
    • Formula:
      • Temperature-Dew Point (°C) divided by 2 = Base of clouds
      • Temperature-Dew Point (°F) divided by 4 = Base of clouds
    • Example:
      • Temperature: 10°C / 50°F
      • Dew Point: 5 °C / 41°F
    • Calculate:
      • (10-5) ÷ 2 = 2,500' MSL

      • (50-41) ÷ 4 = 2,250' MSL


Climb Rate Required

  • Used to determine rate of climb for a given departure/climb out
    • Formula:
      • Ground Speed (GS) (knots) ÷ 60 * Climb Gradient (Feet Per Mile)
    • Example:
      • Ground Speed = 75 knots
      • Climb Gradient Required = 200 feet per mile
    • Calculate:
      • 75 ÷ 60 * 200 = 280 feet per minute climb rate required


Maneuvering Speed:

  • Also referred to Va
  • More weight = more stable
    • Formula:
    • Maneuvering Speed Formula
      Figure 8: Maneuvering Speed Formula





    • Example:
      • Follow instructions given on page 6 of the POH


True Airspeed:

  • The general rule of thumb is to increase TAS by 2% for every 1,000' increase in the altitude
    • Example:
      • Indicated airspeed = 100 knots
      • Altitude = 5,000'
    • Formula:
      • 5 * 0.02 = .1
      • .1 * 100 = 10 knots
      • 100 (IAS) + 10 = 110 knots TAS


Conclusion:


Pilot's Pocket Handbook: Flight Calculations, Weather Decoder, Aviation Acronyms, Charts and Checklists, Pilot Memory Aids
Pilot's rules of thumb: Rules of thumb, easy aviation math, handy formulas, quick tips
Pilot's Pocket Handbook: Flight Calculations, Weather Decoder, Aviation Acronyms, Charts and Checklists, Pilot Memory Aids
Pilot's rules of thumb: Rules of thumb, easy aviation math, handy formulas, quick tips


Maneuvering Speed Formula
Figure 9: Time, Fuel, and Distance to Climb

Top of Climb:

  • Given:
    • Departure Airport: 900 ft
    • Cruise Altitude: 5,500 ft
  • From Sea Level to 5,500' we calculate 9 minutes, 2.0 Gal, 13 NM
  • Assuming 1,000' for the departure altitude we calculate: 1 minute, 0.4 Gal, 2 NM
  • Subtract the difference: (9-1)=8 Min, (2.0-0.4)=1.6 Gal, (13-2)=11 NM
  • Pay attention to the notes at the bottom of the chart, especially to add 1.1 Gal for taxi and takeoff
  • To add wind

References: