General Principles - Take-off — Page 155, Lesson 177

Let’s start with the heart of it: how we actually calculate the take-off distance. I want you to see the two formulae as the skeleton of everything that follows. The upper formula gives the distance required, which we call s, to reach a specified speed V with a given acceleration a. It reads: s = V² / a. So the distance is the square of the speed you need to reach, divided by the acceleration you can produce. That’s why speed matters so much — because it’s squared, doubling the speed quadruples the distance, all else being equal. Beneath that is the formula for the acceleration itself. For an aircraft taking off, the acceleration is thrust minus drag. That’s the net force driving you forward, divided by the mass — but the key point here is that both thrust and drag change as speed changes. So the acceleration is not constant during the take-off. It’s not a nice, steady number you can plug in once and be done with. Also, during the airborne part of the take-off, the laws of motion are somewhat different, so the upper formula changes a bit. But the principle holds: the distance required still depends on the speed to be achieved and the acceleration available. Now, because thrust and drag play such a crucial part, I want to give you a little extra detail on thrust first. The engine thrust will vary during take-off, and the way it varies with speed is different for jet engines and propeller engines. Let’s take the jet engine first. For a jet, the net thrust is the difference between the gross thrust and the intake momentum drag. Gross thrust is the force produced by the exhaust; intake momentum drag is the resistance from accelerating the incoming air up to the aircraft’s speed. As speed increases, intake momentum drag increases, which reduces the thrust. However, at higher speeds, the increased intake pressure due to the ram effect helps to reduce this loss of thrust — and eventually, at very high speeds, it will actually cause the net thrust to increase again. But here’s the important part for take-off: during take-off, the aeroplane speed is still low, so the ram effect is insufficient to counteract the loss of thrust due to intake momentum drag. Therefore, during the take-off, there will be a decrease of thrust. Now, in later chapters and in some performance graphs, you’ll notice the assumption is made that jet thrust is constant with speed. That’s done purely to simplify some teaching points — in reality, it’s falling during take-off. Next, flat rated engines. The thrust produced by an engine at a given rpm depends on the air density, and hence on air pressure and temperature. At a given pressure altitude, decreasing temperature will give increasing thrust. However, many jet engines are “flat rated” — that is, they are restricted to a maximum thrust even though the engine is capable of producing higher thrust. The reason is that at lower temperatures, too much thrust may be generated, and the pressures within the compressors may be exceeded. So, at temperatures below the flat rating cut off — typically about ISA + 15°C — engine thrust is not affected by temperature. Below that point, the engine is held back to its rated maximum, so temperature changes don’t move the thrust. Let me show you the forces at work — — and the net thrust variation with speed for a typical modern jet engine — . And for the flat rating behaviour with temperature — . So, to tie it together: the take-off distance depends on the square of the speed you need and the acceleration you can manage, and that acceleration is thrust minus drag — both of which shift as speed builds. For a jet, thrust actually falls during take-off because intake momentum drag wins over the ram effect at low speeds, and flat rating caps the thrust at low temperatures to protect the compressors. That’s the foundation.