How to Calculate Aircraft Take-Off Ground Run: A Flight Mechanics Tutorial

An aircraft does not become airborne simply because its engines are producing thrust. During the take-off ground run, thrust must overcome aerodynamic drag and rolling resistance. On a sloping runway, it must also overcome a component of the aircraft’s weight.

This tutorial derives a useful ground-run equation from Newton’s second law. It is an engineering model for understanding aircraft performance, not a substitute for approved Aircraft Flight Manual or Pilot’s Operating Handbook data.

What is the take-off ground run?

The ground run is the distance travelled from brake release to lift-off. It is only one part of total take-off distance; the airborne distance needed to clear a specified obstacle is calculated separately.

As the aircraft accelerates, its wings progressively generate more lift. This reduces the load carried by the wheels and therefore reduces rolling resistance. At lift-off, the available lift is sufficient for the aircraft to leave the runway.

For an overview of take-off speeds and phases, including rotation and lift-off speed, see Breaking Down Take-Off: Critical Flight Parameters.

Step 1: Draw the free-body diagram

Start with an aircraft on a level runway and take the forward direction as positive. The important forces are:

• T — thrust, acting forward;
• D — aerodynamic drag, acting opposite to motion;
• R — rolling resistance at the tyres;
• W — weight, equal to mg, acting vertically downward;
• L — aerodynamic lift, acting vertically upward; and
• N — normal reaction from the runway, acting vertically upward.

Step 2: Write the vertical force balance

During the ground run, the aircraft remains in contact with the runway. There is no significant vertical acceleration, so the vertical forces balance:

Therefore:

As lift increases, the normal reaction N falls. The wheels carry less of the aircraft’s weight.

Step 3: Express rolling resistance

Rolling resistance can be modelled as the coefficient of rolling resistance multiplied by the normal reaction:

Substituting N = W − L gives:

Where μ is the coefficient of rolling resistance. This shows why rolling resistance is not perfectly constant during a take-off run. At low speed, lift is small and the wheels carry almost all the aircraft weight. Near lift-off, lift is much greater and rolling resistance is lower.

Step 4: Apply Newton’s second law along the runway

Newton’s second law states that the resultant force equals mass multiplied by acceleration:

For the level-runway ground run:

Substituting the rolling-resistance expression gives the fundamental take-off ground-run equation:

This states that acceleration depends on the force remaining after thrust has overcome drag and rolling resistance.

Step 5: Allow for changing lift and drag

Lift and drag both vary approximately with the square of airspeed:

Where:

• ρ is air density;
• V is airspeed;
• S is wing reference area;
• C_L is lift coefficient; and
• C_D is drag coefficient.

Substituting these terms into the force equation gives:

This is more realistic than assuming fixed acceleration because aerodynamic lift and drag rise as the aircraft accelerates.

Step 6: Convert acceleration into distance

Acceleration is the rate of change of velocity with time:

However, the quantity required is runway distance, s. Using:

The ground-run equation becomes:

Rearranging:

The ground-run distance from rest to lift-off speed V_LO is therefore:

This integral recognises that drag, lift, rolling resistance and potentially thrust change during acceleration.

A simpler constant-acceleration approximation

For an introductory estimate, assume the net accelerating force is approximately constant:

Using the standard kinematic relationship:

If the aircraft starts from rest, U = 0, so:

This approximation is useful for showing trends, but it is not a full aircraft-performance calculation. In reality, drag increases with speed, lift reduces rolling resistance, and available thrust may change as airspeed increases.

Worked example: estimating a ground run

Consider a simplified aircraft model operating from a level, dry runway in still air:

• Mass, m = 6,000 kg
• Weight, W = 6,000 × 9.81 = 58,860 N
• Wing area, S = 30 m²
• Air density, ρ = 1.225 kg/m³
• Take-off thrust, T = 26,000 N
• Rolling-resistance coefficient, μ = 0.02
• Ground-run lift coefficient, C_L = 0.70
• Drag coefficient, C_D = 0.040
• Lift-off speed, V_LO = 60 m/s

First, combine the speed-dependent drag and rolling-resistance terms:

Substituting the values:

The ground-run integral becomes:

Solving gives:

In this deliberately simplified example, the aircraft requires approximately 451 m of ground roll to reach its assumed lift-off speed.

The result does not include rotation, obstacle clearance, wind, runway contamination, engine-transient behaviour, certification margins or the detailed aerodynamic data used in approved performance manuals.

What makes the ground run longer?

Higher aircraft mass

A heavier aircraft increases both W and m. More force is needed to accelerate it, and a higher lift-off speed may also be required.

Higher temperature or altitude

Lower air density reduces lift at a given true airspeed. The aircraft must accelerate to a higher true airspeed to generate the required lift, while engine or propeller performance may also be reduced.

Higher drag

High-lift devices can reduce lift-off speed, but they also alter drag. Engineers must balance the lift benefit against the drag penalty.

A poor runway surface

A higher value of μ increases rolling resistance. Soft, rough or contaminated surfaces can therefore have a substantial effect on acceleration.

Extending the model for runway slope

For an uphill runway with slope angle θ, part of the aircraft’s weight acts down the slope:

The longitudinal equation becomes:

An uphill slope therefore reduces acceleration and increases ground-run distance. On a downhill runway, the weight component acts in the direction of travel and assists acceleration.

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References

1. NASA Glenn Research Center: Four Forces on an Airplane
2. NASA Glenn Research Center: Excess Thrust (Thrust − Drag)
3. Federal Aviation Administration: Pilot’s Handbook of Aeronautical Knowledge