The Velocity Needed for Takeoff calculator computes the velocity required to create more lift than the weight of an aircraft or watercraft using a wing (e.g. hydrofoil).

INSTRUCTIONS: Choose units and enter the following:

• (C_L)  Lift coefficient of the wings.
• (A)  Surface area of the wings.
• (ρ)  Density of the fluid (air 1.2754 kg/m³ at STP or water 998.2071 kg/m³ at STP)
• (M) Mass of the craft.

Takeoff Speed (V):  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

Lift force is created by the flow of a fluid or gas over a wing.  Lift force is a function of the wing characteristics and the velocity of the flow over the wing.  As the velocity increases, so does the lifting force.  At some point, when the velocity is increased, the lift force matches the downward force of gravity on the mass of the vehicle.   That velocity is estimated in this formula.  The Velocity for 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 vehicle's weight.   The formula for the velocity to match force of lift to weight is:

      `V = sqrt((M*g)/(½*A*rho*CL) )`

where:

• V is the velocity (see note)
• M is the Mass of vehicle
• g is the acceleration due to gravity
• A is the wing surface area
• ρ is the density of fluid
• CL is the lift coefficient

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

Note: the wind velocity is not necessarily equal to the speed of the vehicle (e.g. aircraft on the runway).  For aircraft, 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 0.074887lb_m/ft³.

Flight and Navigation Calculators

• Correction Angle: Computes the navigation angle/azimuth correction angle using the wind speed, wind direction, flight heading and an Air Speed.
• Ground Speed: Computes the ground speed based on the wind speed, wind direction, a Flight Heading and an Air Speed.
• Haversine - Distance: Computes the distance between two points on a spherical model of the Earth along a great circle arc.  This also includes the rhumb line distance and azimuth for the rhumb line.
• Travel Time between Coordinates: Computes the time to travel between to points on the globe in a great circle arc at an average velocity. 
• Distance to Sea Level Horizon at Altitude: Computes the distance to the horizon from a specified height using a spherical model the mean spherical radius of the Earth
• Force of Drag: Calculates the resisting force of drag on an object flowing through a medium (e.g. air).
• Force of Lift: Computes the lifting force on the surface area of a wing based on the wing surface area, air flow velocity, density of air and a lift coefficient.
• Lift Coefficient: Compute the lift coefficient of a wing based on lift force, wing surface area, wind speed and density of air.
• Velocity Needed for Takeoff: Computes the velocity required to create more lift than the weight of an aircraft or watercraft using a wing (e.g. hydrofoil).
• Glide Ratio: Computes the glide ratio based on the change in forward distance and the change in altitude.
• Wing Surface Area: Computes the wing surface area required to achieve lift, base on a lift coefficient, lift force, wind speed and density of air.
• Velocity of Air over the Wing: Computes the velocity required to achieve a lift, based on lift coefficient, lift force, wing surface area and density of air.
• Air Speed from Pressures: It uses the Bernoulli Equation to estimate air speed based on the total Pressure measured by a pilot tube, total Static local atmospheric pressure and the Density of Air.
• Center of Mass: Computes the vehicle center of mass between two loads (masses).

References

• Wikipedia - http://en.wikipedia.org/wiki/Density_of_air