TAKE-OFF DISTANCE

Figure 1. Take-off

During ground run, aeroplane accelerates from stop (V = 0) to rotation speed (VR). When rotation speed is achieved, pilot slightly pitches up nose of aircraft and acceleration keeps continue while main landing gears are still on ground, until aeroplane fully lifts-off from the ground at lift-off (VLOF) speed. (Figure 1) Forces acting on the aeroplane during ground run phase are shown in Figure2.

Figure 2. Forces acting on an aeroplane during ground run.

Net accelerating force:

In this equation μ(W-L) is the friction force between runway and aeroplane’s wheels. Friction coefficient is usually assumed to be μ=0.02 for concrete runways. From Newton’s second law:

where m = W/g is the mass of aeroplane and a = dV/dt is acceleration of the aeroplane. If horizontal distance travelled on ground is represented by Xg

Therefore:

Remember that

and

In this case:

If it is assumed that the thrust is constant during ground run, it is easy to find a solution of above integral equation by assuming

and

Thus

Ground run becomes

or

Lift off speed is usually 10% higher than the stalling speed. Since

then

Therefore, ground run

During airborne phase, aeroplane climbs to screen height (H) defined by regulations while accelerating from lift off speed to take off speed of V2. Regulations define the screen height 35 ft for Class A aeroplanes and 50 ft for Class B aeroplanes. V2 is defined 20% higher than the stalling speed for both classes of aeroplanes. Forces acting on the aeroplane during airborne phase are shown in Figure 3.

Figure 3. Take-off climb

Although aeroplane accelerates from VLOF to V2, we can assume that it climbs with a constant speed which is average of these two speeds. Thus

Climb angle

For Class A and B aeroplanes, usually initial climb takes place with a small angle (g < 15°), thus

Since aeroplane climbs to screen height of H with angle g, then airborne distance

For small angles

Therefore

or

or

The total take-off distance becomes

This equation presents influence of various factors on take-off distance:

1 – Higher take-off mass, which means higher take-off weight, results in longer take-off distance.
2 – Higher thrust-to-weight ratio (T/W) results in shorter take-off distance.
3 – Lower air density, which means higher temperature or higher runway elevation, results in longer take-off distance. Lower density also causes lower thrust, which leads to further extension of take-off distance.
4 – Higher maximum lift coefficient up to a certain extent can lead to a shorter take-off distance. However, higher maximum lift coefficients may cause higher drag coefficients for some flap settings, and this can result in longer take-off distance.
5 – For similar thrust-to-weight (T/W) and ratios, airborne distances of Class B aeroplanes are always longer compared to Class A aeroplanes.

Figure 4. Lufthansa A-319 at about screen height during take-off.

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