Aircraft Speed - Lift - Weight

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The Aircraft Speed Needed for Takeoff calculator computes the speed required to create more lift than the weight of the aircraft.

INSTRUCTIONS: Choose units and enter the following:

• (CL)  This is the lift coefficient of the wings.
• (A)  This is the surface area of the wings.
• (ρ)  This is the density of the air.
• (W) This is the weight or mass of the aircraft.

Takeoff Speed (TS):  The calculator returns the required velocity (speed) in meters per second.  However, this can be automatically converted to compatible units (e.g. miles per hour) via the pull-down menu.

The Math / Science

The Aircraft Speed (lift to overcome weight) formula computes the speed needed to achieve the lifting force on the surface area of a wing that is greater than the aircraft's weight.   The formula for the force of lift is:airframe forces

L = ½•A•ρ•CL•V²

where:

• L is the force of lift
• A is the wing surface area
• ρ is the density of fluid
• CL is the lift coefficient
• V is the velocity of flow

Flight can be achieved with the force of lift is greater than the weight.

The algorithm uses a wind velocity starting with 0.1 m/s and increases the value until the lifting force exceeds the weight of the aircraft.  The formula returns that velocity. Note: the wind velocity is not necessarily equal to the speed of the aircraft on the runway.  Tail and head winds contribute to the total wind speed flowing over the wings.  For precisely this reason, aircraft carriers turn into the wind to add both the ship's velocity and the speed of the wind to velocity of the aircraft being launched.

Science of Lift

Bernoulli's principle is based on the conservation of energy, which dictates that in a steady flow of a fluid (lacking any substantial turbulence) the sum of all mechanical energy along a line of flow, a streamline, is the same at all points on that flow path. This , in turn means the sum of the potential and kinetic energy must remain constant and so with increased velocity of the flow, there is an decrease in static pressure.

From this same Bernoulli's principle we can derive the equation to calculate the lift force on a wing surface (airfoil). When the air flowing past the top surface of an aircraft wing moves faster than the air flowing past the bottom surface, Bernoulli's principle defines a difference in pressure on the two surfaces of the wing, with the lower pressure being on the upper surface where the faster flow exists.

The difference in pressure sums to a net upwards lifting force, as calculated in this equation..

Density of Air

The density of air, ρ (Greek: rho) (air density), is the mass per unit volume of Earth's atmosphere. Air density, like air pressure, decreases with increasing altitude. It also changes with variation in temperature or humidity. At sea level and at 15 °C, air has a density of approximately 1.225 kg/m3 (0.001225 g/cm3, 0.0023769 slug/ft3, 0.0765 lbm/ft3) according to ISA (International Standard Atmosphere).

• At IUPAC standard temperature and pressure (0 °C and 100 kPa), dry air has a density of 1.2754 kg/m3.
• At 20 °C and 101.325 kPa, dry air has a density of 1.2041 kg/m3.
• At 70 °F and 14.696 psi, dry air has a density of 0.074887lbm/ft3.