Modern Lift Equation Calculator

The modern lift equation shows that the lift of a wing depends on four factors: air density, speed of the airflow around the wing, wing area and the lift coefficient which describes both the shape of the airfoil and its angle of attack. This is the equation that aircraft designers use first thing in the morning, and it also explains why a wing that looks light and thin on the ground can support several tons of weight in the air. If you specify four out of five values, then the fifth one can be calculated.

What the modern lift equation means:

The actual lift depends on a long list of factors including air density, the square of velocity, viscosity and compressibility of the air, wing area, shape of the airfoil, and angle between the wing and airflow. Most of these factors are difficult to determine individually.

The essence of the modern equation is to summarize all effects of shape, angle of attack and flow into a single measured number, called the lift coefficient $C_L$. This turns a complex problem into a simple multiplication. The lift is half the air density times the square of the speed times the wing area, multiplied by this one coefficient again.

Here, $L$ is the lift force, $\rho$ is the air density, $v$ is the relative velocity, $A$ is the wing area and $C_L$ is the lift coefficient. The same relationship can also be written as $L = C_L\,A\,\tfrac{1}{2}\rho v^{2}$, where the last term represents the dynamic pressure carried by the flowing air.

Why is it called a modern lift equation?

Earlier explanations of lift were based on single mechanisms such as the assumption that air takes longer to travel over the top of a wing than the bottom, leading to incorrect values. Modern methods do not attempt to derive lift from a simple explanation.

Instead it is acknowledged that shape, angle of attack and flow conditions interact in a complex way, all summarized in an "lift coefficient" $C_L$. This can be measured in a wind tunnel or computed by simulation. So this equation is modern and reliable. It faithfully summarizes the situation in numerical values, rather than hide complicated physical laws in a neat but incomplete explanation.

Use of the calculation tool:

This calculator will return the lift given the air density, velocity, wing area and coefficient of lift. Conversely, you can leave the desired field blank and enter the other four values to calculate the corresponding quantity. No mode selection is required; the calculation algorithm will determine which quantity is missing based on which field you left blank.

This covers everyday tasks such as determining the wing size for a given weight, finding a speed that produces enough lift or calculating the lift coefficient from a known amount of lift. If you enable "Angle of Attack" option then you can predict the coefficient based on the angle of attack. By enabling "Minimum Flight Speed" option and entering a value, you can calculate the stall speed.

Examples of calculations:

Let's take a small general aviation plane as an example. The wing area is 15 m², the speed is 50 m/s (about 180 km/h), and the air density at sea level is 1.225 kg/m³. We'll assume a lift coefficient of 1.2 for cruising flight. If we plug these values into the equation, we get the following:

This corresponds to a lift of about 2810 kgf which is enough to support the airplane in level flight. If the speed is increased to 100 m/s, then the lift increases to about 110250 N. As the velocity factor is squared, this value quadruples. The following table shows the values for each symbol in this example.

Rearranging Equations to Solve for Any Variable

The lift equation is a simple product and therefore can be easily rearranged. This calculator uses the following format and will select the appropriate format based on which fields you leave blank.

The first format calculates the required lift coefficient for a wing, given a known lift and speed. The second format shows the required speed to carry a certain weight, and is thus central to take-off calculations. The third format determines the required wing area based on desired lift and chosen cruise speed.

What is dynamic pressure?

The value of $q = \tfrac{1}{2}\rho v^{2}$ represents the dynamic pressure. This is the pressure exerted by air as it flows due to its motion. It is the actual driving force behind lift and drag and corresponds to the value that appears in the Bernoulli equation.

With the dynamic pressure, the lift equation becomes $L = q\,A\,C_L$. This calculation tool outputs both the dynamic pressure $q$ and average lift pressure $L/A = q\,C_L$ separately. The latter is the net difference of positive upward pressure between the bottom and top side of the wing averaged over the surface area of the wing.

The lift coefficient is crucial.

The lift coefficient is a dimensionless number that summarizes the shape of an airfoil, its angle of attack and flow conditions into one value. For a thin symmetric airfoil at zero angle of attack, it is close to zero. As the angle of attack increases, the lift coefficient increases until it reaches a critical point.

When the critical angle of attack is exceeded, flow separates from the top surface of the wing, the coefficient suddenly drops and the wing stalls. This upper limit defines the maximum lift that a given airfoil can produce at a constant speed. Typical values are listed below as a starting point.

Lift curve as a function of angle of attack.

Before stall, the lift coefficient increases almost linearly with angle of attack. This linear region is called the lift curve, and its slope is determined by the airfoil shape. Thin airfoil theory predicts a slope of $2\pi$ per radian, or about 0.11 per degree for a two-dimensional section. The actual slope for a finite wing is somewhat lower.

Here $a_0$ is the lift curve slope, $\alpha$ is the angle of attack and $\alpha_0$ is the zero-lift angle of attack. For a symmetrical airfoil this value will be 0 while for a cambered wing it will be slightly negative. By activating the "Angle of Attack" option these values can be calculated to check the resulting lift against the inputted values. For an airfoil with a slope of 2$\pi$ at an angle of attack of 5 degrees this value would be about 0.55.

Air density and altitude

Air density changes dramatically with altitude and temperature, which directly affects lift. At sea level in a standard atmosphere the air has a density of 1.225 kg/m$^3$. At about 3,000 m the density is down to around 0.9. In cruise flight at passenger jet altitudes it's about one third of that value at sea level.

The thinner the air, the less lift is produced at a given speed. So a wing produces less lift high in the mountains or on a hot day than it does on a cold morning at sea level. Pilots call this density altitude. For that reason, a fully loaded airplane taking off from a mountain airport on a hot day needs a longer runway.

Minimum flight speed and stall speed

If you enable the minimum flight speed option and add the weight that must be carried by the wings, then you can determine the lowest speed at which an aircraft can fly. For a stable horizontal flight, lift must equal weight. If we denote the speed as $L = m g$, then the following equation is obtained:

If the flight speed falls below this value, the wing will not generate enough lift and stall. This value is close to the stalling speed and take-off speed and is calculated based on the entered lift coefficient. The calculation tool also outputs the wing loading which is obtained by dividing the weight by the wing area. This is an indicator for a quick estimation of the load on the wing.

Application of the lift equation

This equation plays a significant role in various engineering disciplines though often behind the scenes. Aircraft designers use it to determine the minimum size of a wing that can lift a given weight at a particular speed. Test flight teams measure the lift coefficient at different angles of attack and plot performance curves from which pilots fly the aircraft.

Racing engineers use this equation in reverse to design wings that push the car down instead of up, making the coefficient negative. The same physics is found in wind turbine blades, determining the size of drone rotors, underwater foils that lift a ship's hull (hydrofoils), and even the flight path of a kicked soccer ball.

Tips for a good estimate

Choose a wing area that corresponds to the desired lift coefficient. Published values are for some reference area, usually the planform area of the wing. Use an air density appropriate for altitude and temperature, and do not always assume sea level values.

If accuracy is important, determine the lift coefficient from a database of airfoil profiles or wind tunnel measurements. The lift coefficient varies with angle of attack and airfoil profile. The velocity must be specified as relative velocity, i.e., the velocity relative to the air. This is because the actual airflow over the aircraft is affected by headwinds or tailwinds.