Physics question
Suppose I can ride at 20 MPH w/o any wind. How fast should I be able to ride into a 20 MPH headwind? With a 20 MPH tailwind?
Thanks! 
How much do you weight and what bike are you on?

Are we assuming that you mean you can sustain 20 MPH with no wind, or is that top speed?
The forces limiting your speed are: 1) air resistance 2) tire rolling resistance 3) mechanical friction At speeds above around 15 mph, the second two become less and less important as your speed increases. Air resistance increases exponentially as the combined sum of your speed + the wind speed increases, while the other two increase more or less linearly with your speed. So, if you ride at 1 MPH into a 20 MPH headwind, the force you must overcome from air resistance is the same as the force of riding 21 MPH with no wind, but the mechanical friction and rolling resistance would be about 1/20th as much. This difference determines how fast you can go. It won't be very fast. With a 20 MPH tailwind, the air resistance from riding at 20 MPH is nearly canceled (though not exactly), so you can go much faster. Still you won't be able to go 40 MPH because the other two forces are still increasing as you gain speed. All of this assumes we're talking about riding on flat surfaces. Gravity, of course, throws in additional wrinkles. There are also undoubtedly psychological forces that come into play as riding into a headwind can very quickly drain your will to live. 
It's a trick question. The answer is, of course, not fast enough.

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20 MPH minus whatever your profile (parasitic) drag increase is for the added wind.

There are some online power calculators. Hunt up one of them, check your power output in the first situation, then see what speed that gives you in the other situations.

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If the air is still, when you ride 20 mph, you are going to have an effective 20 mph headwind. However, if you are riding into a 10 mph wind at a constant speed of 10 mph, the head wind you experience varies. If you get in the lee of a large object, the head wind will drop. Hills can shelter you from the wind. Also as you approach ground level, the effective wind speed will drop because it will be blocked by even relatively small objects like tall grass, fences, parked cars and so on. So when you are riding into a constant 10 mph headwind at 10 mph, the very biggest effective headwind you can get is 20 mph. That is probably the effect you are noticing. 
Here's what I know for sure:
With a barely perceptable tailwind I can ride out, away from my car, and convince myself that I'm riding fast because I'm in better shape than I thought. Eventually I have to turn around to ride back.... You know the rest. 
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Also, spokes do a good bit of air churning no matter what the wind speed. 
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