Ted wrote:
While the aircraft is on the ground it is subject to a lateral force that must be balanced in some way, this would create drag, as soon as the aircraft leaves the ground, there is an unbalanced lateral force that will accelerate the aircraft in some way. Even if you do nothing the aircraft will yaw by itself and there should be an increase in airspeed unless all of the impulse is absorbed by the yaw.
Why would airspeed increase? The impulse is at 90degrees to the aircraft and would accelerate it east in my diagram.
To also respond to other posts, the only relative wind that matters for “take off distance” is the relative wind along the longitudinal axis. The plane will never take off on the ground from a strong crosswind (well, it can be blown off but that’s an entirely different thing)
And as to ted’s comment, the wind blowing the plane to the side right after take off, the ground speed will increase (just look at Jason’s diagram) but Performance (/ASI) will remain the same, as longitudinal is what matters.
I’m also discarding AnthonyQ very valid point that deflected controls to counteract wind will actually decrease performance compared to a no crosswind situation
I think the answer to this puzzle is one which cannot be disposed of with one-liners about wind triangles. It needs a bit of time spent on typing and drawing some diagrams. One aspect is that the plane is not flying while on the runway. Another is what assumption is made about the acceleration versus what the air around the plane is doing (this assumption is probably that the acceleration is, to a first order approximation, prior to Vr, independent of the wind, because, especially with a CS prop, the engine power is being mostly converted into accelerating the mass). Another is the extent to which the wing lift is sensitive to the deviation from perpendicular of the airflow across it (and this will be different for the two wings, with the wing which is “nearer” the crosswind getting more lift, and the other getting less lift due to fuselage shielding, which is obviously why you have to use ailerons to keep the plane flat, prior to Vr; the rudder alone doesn’t do it).
Peter, I think that is what everyone is trying to discuss. In the OP you said: “flying in a crosswind creates a headwind and the faster you go the more headwind it makes for you.”
Could you explain that comment?
But I just don’t think it is that much of a puzzle. The question is about takeoff run. What makes the plane generate enough lift for takeoff is airspeed over the wing. A crosswind just can’t affect that.
JasonC wrote:
Why would airspeed increase? The impulse is at 90degrees to the aircraft and would accelerate it east in my diagram.
Indeed it would be displaced to the east and relative to the ground it would have accelerated in that direction. However the aircraft’s lateral momentum relative to the centreline should resist this lateral acceleration, and unless your flying a brick it will result in a yaw and an increase in airspeed.
I think this point is largely academic, however I think that is point of Peter’s puzzle 
Ted wrote:
Indeed it would be displaced to the east and relative to the ground it would have accelerated in that direction. However the aircraft’s lateral momentum relative to the centreline should resist this lateral acceleration, and unless your flying a brick it will result in a yaw and an increase in airspeed.
Ok I accept the inertia will lead to a small airspeed increase temporarily while the aircraft is accelerated east by the relative airflow. But lets consider the magnitude. In my wind vector with a 20kt crosswind at a 60 knot rotation speed you get a resultant vector of 63 knots. And that is only a temporary increase. It also doesn’t help you lift off, only when airborne.
AeroPlus wrote:
Peter is right.
Not entirely. Think of a constant cross wind 90 deg from the left. The faster you go, the more the relative wind comes straight on (0 deg), but the wind component is still constant and coming 90 degree from the left. You have to change heading into the wind to take advantage of it, as headwind. Following the center line all the way, only a helicopter would (theoretically, but rather meaningless) be able to utilize this, yawing constantly directly into the relative wind all from the start (starting 90 deg relative the center line).
An airplane would have to move diagonally across the runway to utilize a bit of the cross wind.
JasonC wrote:
Once you are airborne in the airmass a turn into the crosswind will affect angle of climb due to lower ground speed but won’t increase wind over the wings.
Well, the point is, you are already moving against the wind (considering the wind is 90 degrees, and you are moving straight down the runway). Only the wind component is unusable to you. Therefore turning into the wind the moment you are airborne will increase the airspeed. But, you have to turn faster than it takes for the aircraft to become fully “blown away” and emerged in the air masses so to speak. You also have to keep the momentum fwd relative to the aircraft, not simply yaw the aircraft for full effect.
Take off speed is 50 kt, and the wind is coming 90 degrees from the left at 50 kt. At t0 all you have is wind coming from the left. At t1 your wheels leave the ground and you are moving 50kt across the ground, and your head wind component is 50 kts. But the relative wind component is sqrt(50² + 50²)=70.7 knots at 45 degrees. Now, just turning (yawing) into the relative wind and you will have the airspeed of 70.7 kt in an instant. What you are doing is straightening up from a rather insane side slip. Now, instead of just yawing, you are keeping the side slip and turning the aircraft straight into the wind while keeping the fwd momentum. At the end you will have and airspeed of 100 knots going straight into the wind.
Being able to do that without everything just dissipating in extra drag is of course not very likely, but some of the effect should be perfectly obtainable (of course with a bit less than 50 kt cross wind )
Wind is air in motion and does not have any effect (apart from gusts and disturbances possibly) on air speed, lift, stability, and control: an airplane, once in flight, cannot “feel” the wind.
On a cross-wind take-off, right after you break ground, you can for just a few seconds feel the sideways shoving of the air, shouldering you over, accompanied by a weathercocking tendency. However, this lasts just seconds and once the airplane has yielded to this force, it is flying and cannot feel the wind. This shoving tendancy right after take-off has to do because of the force that just was acting on it.
On the runway, I would agree that the sidewind has no or hardly any effect on the required take-off distance.
I would not know how your airspeed could suddenly increase once airborne.
Here is my take on it:
The main factor is that the plane is not flying while on the runway i.e. prior to Vr. So no point in looking at wind triangles during the takeoff run.
A reasonable assumption is that the acceleration is, to a first order approximation, and prior to Vr, independent of what the air is doing. This is because, especially with a CS prop, the engine power is being mostly converted into accelerating the mass, and we can consider that to be the case until Vr (liftoff).
At Vr, say 70kt, and with a xw of say 20kt, you have the following
So you have 73kt of airflow over the wings, and there is no question about that. You do “make your own wind” by moving in a crosswind; one learns this very quickly when e.g. windsurfing. The more rapid buildup in IAS is also blindingly obvious on a xw departure in the TB20, and it is real, not just an artefact of the pitot tube (which would in any case show a smaller reading for any off-axis airflow). This is also why the average GS calculated over any large number of flights is always less than the TAS – because the “average wind” (the wind vector calculated over some large number of flights) is an exact crosswind except for the impossible scenario of always flying in calm air, yet even an exact crosswind gives you a headwind when you are flying. This is well known. Wind always, on average, costs you speed – because, wind, always, on average, means you have a headwind.
One might argue that the increased headwind will reduce the acceleration, but that merely supports the assertion that the headwind is useful 
But it is 16 degrees off axis. The Q is: how much does this matter? How much is the wing lift sensitive to the deviation from perpendicular of the airflow across it? The lift will also be different for the two wings, with the RH wing getting more lift than the LH wing, due to fuselage shielding of the LH one. This is obviously why you have to use ailerons to keep the plane flat, prior to Vr; the rudder alone doesn’t do it. If you don’t use into-wind ailerons, the RH wing will start to lift sooner.
I would assert that at 16 degrees the extra 3kt will be mostly beneficial. The situation is monotonic, so the only Q is “how much benefit”. There is some extra drag due to the ailerons and the rudder but this starts at zero and builds up, whereas the engine power is being converted into speed continuously. Especially at these low speeds, the ailerons also don’t drag that much; if they did, you would have a huge need to use the rudder when turning which (on a SEP tourer; gliders are different) you don’t. The ailerons are reasonable aerofoils.
Is 3kt relevant? Well, look at the huge increase in MTOW etc which the TBM and the PC12 got by raising the mandatory max Vs (for any SE plane) from 60kt to 65kt.
After liftoff (Vr) the picture changes. Over a few seconds, the plane turns into the relative wind (i.e. a heading change from 360 to 016 in this case). The wind also gets stronger because you are climbing into a wind shear (wind gradient) and the aircraft mass means some kinetic energy gets traded for a higher rate of climb (a faster acquisition of potential energy)… for a short time. That’s why a departure into wind gives you a briefly higher rate of climb. The ROC also varies according to whether you turn right or left straight away; if you turn right you get a higher brief ROC because the wind gradient is stronger. Also the wind direction turns right (CW) as you are higher up, which also favours a right turn. BUT this whole paragraph is irrelevant after you lift off because the runway length is now immaterial.
All very long winded But the lift at Vr is calculated from the 70kt component at zero degrees…which is the same as the nil wind case.
