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Minimum total drag occurs where the two drag curves cross

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baddles created the topic: Minimum total drag occurs where the two drag curves cross

Hi all, just wanted to share an insight about minimum drag.

We're used to seeing the total drag curve, which is the sum of two curves, the induced drag and parasite drag, like this:

The textbooks all say that the minimum value of total drag is achieved where the two drag curves intersect, that is, where induced drag equals parasite drag. The picture seems to confirm it.

But this kind of statement is not true in general - if you have any old two curves, one decreasing and one increasing, then the total of the two curves is usually not minimised where the curves cross. Here is a counterexample, in which the downward-sloping green curve is y = 16/x and the upward sloping blue curve is y = x2.

This was driving me crazy - until I realised that the drag curve is a special case. The parasite drag is proportional to V2 and the induced drag is proportional to 1/V2 where V is the IAS. With a bit of calculus, it turns out that the minimum is achieved where the curves cross.

The rule that "the minimum happens where the curves cross" is only true because of the special shapes of these curves.

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