Calculating v Speeda at Different Weight

Calculate Stall Speed at Varying Aircraft Weights

Calculator Inputs

Key Intermediate Values

Weight Ratio: —

Adjusted V_s (Proportional): —

Final V_s: —

Key Assumptions

Weight Units: —

Speed Units: —

Formula Used: V_{s_new} = V_{s_ref} * sqrt(Weight_new / Weight_ref)

Formula Explanation: The stall speed (V_s) of an aircraft is directly proportional to the square root of the wing loading (which is weight divided by wing area). Assuming the wing area remains constant, stall speed is proportional to the square root of the aircraft's weight. This formula allows us to estimate the stall speed at a different weight by scaling the known stall speed at a reference weight.

Stall Speed vs. Aircraft Weight

V_s Calculation Details

Input	Value	Unit
Reference Weight	—	—
Reference Vs	—	Knots
Current Weight	—	—
Weight Ratio (Current/Reference)	—	—
Adjusted Vs (Proportional)	—	Knots
Final Calculated Vs	—	—

What is V_s (Stall Speed)?

V_s (Stall Speed) is a critical aerodynamic parameter for any aircraft. It represents the minimum speed at which the aircraft can maintain controlled flight. Below V_s, the wings are no longer generating enough lift to counteract the aircraft's weight, leading to a stall – a sudden loss of lift and an uncontrolled descent. For pilots, understanding and respecting V_s is paramount for safe operation, especially during takeoff, landing, and low-speed maneuvering. It's typically quoted in knots (kts), but can also be expressed in miles per hour (mph) or kilometers per hour (kph).

This value isn't static; it changes based on several factors, with aircraft weight being one of the most significant. Heavier aircraft require more lift to stay airborne, which in turn necessitates a higher airspeed to achieve that lift. Therefore, as an aircraft gets heavier, its V_s increases. Conversely, a lighter aircraft will have a lower V_s. Pilots must constantly be aware of the aircraft's current weight and its corresponding V_s to ensure they operate within safe flight parameters.

Who should use this calculator? This V_s speed calculator is invaluable for pilots (student and experienced), flight instructors, aviation enthusiasts, and aircraft designers. It helps in flight planning, understanding aircraft performance limits, and appreciating the direct relationship between weight and the minimum safe flying speed.

Common Misconceptions:

• V_s is a fixed number: Many believe V_s is a single, constant value for an aircraft. In reality, it varies significantly with weight, altitude, configuration (flaps, gear), and even air density.
• Stall happens suddenly at any speed: While a stall is a sudden loss of lift, it occurs specifically when the critical angle of attack is exceeded, regardless of the airspeed, provided the airspeed is below V_s for the current conditions.
• More weight means you can fly faster: The opposite is true. More weight requires more lift, and thus a higher speed to achieve that lift, increasing V_s.

V_s Formula and Mathematical Explanation

The relationship between an aircraft's weight and its stall speed (V_s) can be understood through the principles of aerodynamics, specifically lift generation. The lift (L) generated by an aircraft's wings is given by the formula:

L = ½ * ρ * V² * S * C_Lmax

Where:

• L is the lift force
• ρ (rho) is the air density
• V is the airspeed
• S is the wing surface area
• C_Lmax is the maximum coefficient of lift

In level, unaccelerated flight, lift (L) must equal the aircraft's weight (W). At the point of a stall, the wings are producing maximum lift, so we use C_Lmax. Therefore, at stall speed (V_s), we have:

W = ½ * ρ * V_s² * S * C_Lmax

Rearranging this formula to solve for V_s:

V_s = √( (2 * W) / (ρ * S * C_Lmax) )

This shows that V_s is proportional to the square root of the weight (W), assuming air density (ρ), wing area (S), and C_Lmax remain constant.

Our calculator utilizes a simplified, practical version of this principle. It compares the current weight to a reference weight and scales the reference stall speed accordingly:

V_{s_new} = V_{s_ref} * √(Weight_new / Weight_ref)

This formula allows us to estimate the stall speed at a new weight based on a known stall speed at a reference weight.

Variable Explanations

V_s Calculation Variables

Variable	Meaning	Unit	Typical Range
Vs_ref	Reference Stall Speed	Knots, mph, kph	30 – 100+ kts (depends heavily on aircraft type)
Weightref	Reference Weight (e.g., Max Gross Weight)	lbs, kg, N	Varies widely by aircraft
Weightnew	Current or New Aircraft Weight	lbs, kg, N	From empty weight to max gross weight
Vs_new	Calculated New Stall Speed	Knots, mph, kph	Adjusted based on weight ratio
Weight Ratio	Ratio of New Weight to Reference Weight	Unitless	0.1 – 2.0 (typically)
sqrt(Weight Ratio)	Square root of the weight ratio	Unitless	0.3 – 1.4 (typically)

Practical Examples (Real-World Use Cases)

Example 1: Light Training Aircraft (e.g., Cessna 172)

A Cessna 172 has a maximum gross weight (Weight_ref) of 2550 lbs. Its stall speed in the clean configuration (landing gear up, flaps up) at this weight (V_{s_ref}) is approximately 55 knots.

Scenario: A pilot is flying the Cessna 172 with passengers and full fuel, bringing the current weight (Weight_new) to 2400 lbs. They want to know the approximate stall speed during the approach.

Inputs:

• Reference Weight: 2550 lbs
• Reference V_s: 55 kts
• Current Weight: 2400 lbs
• Weight Units: lbs
• Speed Units: kts

Calculation:

• Weight Ratio = 2400 lbs / 2550 lbs ≈ 0.941
• sqrt(Weight Ratio) ≈ sqrt(0.941) ≈ 0.970
• New V_s = 55 kts * 0.970 ≈ 53.35 kts

Result: The estimated stall speed (V_{s_new}) for the Cessna 172 at 2400 lbs is approximately 53.4 knots.

Interpretation: Because the aircraft is lighter than its maximum gross weight, the stall speed has decreased slightly. The pilot should maintain an airspeed slightly above 53.4 kts to avoid a stall during landing.

Example 2: Heavier Utility Aircraft (e.g., Piper PA-31 Navajo)

A Piper PA-31 Navajo has a maximum gross weight (Weight_ref) of 8000 lbs. Its stall speed in landing configuration (gear down, flaps down) at this weight (V_{s_ref}) is approximately 72 knots.

Scenario: The same aircraft is configured for takeoff, but due to payload limitations, its weight is only 7000 lbs (Weight_new). The pilot needs to estimate the takeoff stall speed (in the clean configuration, let's assume V_{s_ref} at 72 kts is a close approximation for this purpose, though ideally, a configuration-specific value would be used).

Inputs:

• Reference Weight: 8000 lbs
• Reference V_s: 72 kts
• Current Weight: 7000 lbs
• Weight Units: lbs
• Speed Units: kts

Calculation:

• Weight Ratio = 7000 lbs / 8000 lbs = 0.875
• sqrt(Weight Ratio) = sqrt(0.875) ≈ 0.935
• New V_s = 72 kts * 0.935 ≈ 67.32 kts

Result: The estimated stall speed (V_{s_new}) for the Piper PA-31 at 7000 lbs is approximately 67.3 knots.

Interpretation: With a lighter load, the Navajo's stall speed is reduced. This means the aircraft can potentially take off at a lower indicated airspeed compared to operating at its maximum gross weight, which is a beneficial performance characteristic.

How to Use This V_s Calculator

1. Enter Reference Weight: Input the known maximum or standard gross weight of your aircraft. This is the weight at which the reference stall speed was determined.
2. Enter Reference V_s: Input the stall speed (V_s) that corresponds to the reference weight. Ensure you know the units (knots, mph, or kph).
3. Enter Current Aircraft Weight: Input the actual weight of the aircraft for the flight condition you are analyzing (e.g., takeoff weight, landing weight).
4. Select Weight Units: Choose the unit of measurement (lbs, kg, N) that matches your entered weights.
5. Select Speed Units: Choose the desired unit (knots, mph, kph) for the calculated stall speed output.
6. Click "Calculate V_s": The calculator will process the inputs.

Reading the Results:

• Calculated Stall Speed (V_s): This is the primary output, showing the estimated stall speed for your current aircraft weight and selected speed units.
• Key Intermediate Values: These provide insight into the calculation steps: the ratio of the current weight to the reference weight, and the proportional adjustment factor.
• Key Assumptions: This section confirms the units used and the underlying formula.
• Table: A detailed breakdown of all input values and calculated results in a structured format.
• Chart: A visual representation showing how stall speed changes across a range of weights, based on your inputs.

Decision-Making Guidance: Always maintain a safety margin above the calculated V_s. Regulatory requirements and aircraft POH (Pilot's Operating Handbook) often specify minimum approach speeds or margins (e.g., V_ref, final approach speed) which are higher than V_s. This calculator helps you understand the *minimum* speed for potential lift loss, informing your approach speed planning. For instance, if your calculated V_s is 55 kts, your final approach speed might be set at 70-80 kts or more, depending on aircraft type and conditions.

Key Factors That Affect V_s Results

While our calculator focuses on weight, it's crucial to understand that real-world V_s is influenced by many factors. Here are some key ones:

1. Aircraft Configuration: The deployment of flaps, slats, and landing gear significantly alters the wing's lift characteristics. Flaps increase lift (allowing for lower V_s) but also increase drag. Our calculator assumes V_{s_ref} and V_{s_new} are in the same configuration (typically 'clean' for takeoff calculations or 'full flaps' for landing).
2. Air Density (Altitude & Temperature): Higher altitudes and higher temperatures mean less dense air. Less dense air requires a higher True Airspeed (TAS) to generate the same amount of lift as in denser air at lower altitudes. Our calculator implicitly assumes similar air density for both reference and current conditions, or that V_{s_ref} was determined at a comparable density. In reality, V_s increases with altitude.
3. Angle of Attack (AoA): V_s is the speed at which the critical AoA is reached. Exceeding this AoA, regardless of speed, causes a stall. Factors like turbulence or improper control inputs can lead to a stall even when flying above the calculated V_s.
4. Load Factor (G-Force): Maneuvering an aircraft, especially pulling up or turning sharply, increases the effective weight experienced by the wings (load factor). This increased effective weight requires more lift, thus increasing the stall speed. Our calculator assumes a load factor of 1 G (level flight). Aggressive maneuvers can significantly raise the stall speed.
5. Wing Contamination: Ice, frost, or even heavy rain on the wings can disrupt airflow, increase drag, and crucially, reduce the maximum lift coefficient (C_Lmax). This directly increases the stall speed, often dramatically. Flying with contaminated surfaces is extremely dangerous.
6. Center of Gravity (CG): While weight is the primary factor, the CG position affects the aircraft's stability and the pilot's ability to achieve and maintain optimal angles of attack. A CG closer to the forward limit might slightly alter handling characteristics and potentially V_s compared to a CG at the aft limit, although the impact is less direct than total weight.
7. Turns: In a standard rate turn (3 degrees per second), the load factor is approximately 1.25 G. This alone increases stall speed by about 10%. Steeper turns increase the load factor and stall speed even more dramatically.

Frequently Asked Questions (FAQ)

Q1: Does this calculator provide the exact stall speed from the aircraft manual?

A: This calculator provides an *estimate* based on the V_s formula relating weight and stall speed. The official V_s in your Pilot's Operating Handbook (POH) accounts for specific configurations, air density, and potentially CG. Always refer to your POH for definitive values.

Q2: Why is my calculated V_s different from what I see in the POH?

The POH V_s is usually for a specific configuration (e.g., flaps up, full flaps) and maximum gross weight. Our calculator allows you to adjust the weight and assumes the same configuration for both reference and current states. Differences can also arise from air density assumptions.

Q3: Can I use this calculator for takeoff stall speed?

Yes, if you use the reference V_s and weight corresponding to the clean configuration (flaps retracted) and enter the takeoff weight. Remember to use a safety margin above the calculated V_s for takeoff.

Q4: How does landing configuration affect stall speed?

Landing configurations (flaps extended, landing gear down) generally increase lift, allowing the aircraft to fly slower before stalling. This means V_s in landing configuration is typically lower than in clean configuration at the same weight. If your reference V_s is for landing configuration, ensure your current weight calculation also reflects landing configuration assumptions.

Q5: What does a "Weight Ratio" of 1.5 mean?

A Weight Ratio of 1.5 means your current aircraft weight is 1.5 times (or 50% heavier than) the reference weight you entered. This will result in a significantly higher stall speed.

Q6: How much safety margin should I add to the calculated V_s?

This depends on the phase of flight and aircraft type. For final approach, speeds are typically 1.3 times V_s (or higher, depending on POH recommendations) plus adjustments for wind. For takeoff, ensure you are well above V_s before lifting off. Always consult your aircraft's POH for specific speed requirements.

Q7: Does air density affect stall speed?

Yes, significantly. At higher altitudes or temperatures (less dense air), the aircraft needs to fly faster (higher TAS) to generate the same amount of lift. Therefore, stall speed increases with altitude and temperature, even if weight remains constant. This calculator simplifies by assuming similar air density conditions.

Q8: What is the 'Critical Angle of Attack'?

The critical angle of attack (AoA) is the specific angle between the wing's chord line and the oncoming relative wind at which the wing stalls. This angle is relatively constant for a given airfoil (around 15-18 degrees for many conventional airfoils), regardless of airspeed or weight. V_s is the speed at which this critical AoA is reached at maximum lift coefficient (C_Lmax).

s

s

Weight Ratio:

Adjusted V_s (Proportional):

Final V_s:

Weight Units:

Speed Units: