How fast do you need to be going to get an effective slipstream on a bike?
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It depends. I don't know of any scientific research to support an actual speed.
It is not necessarily how fast you go, but how closely you are following the person cycling in front of you. The closer you follow, the more likely you are to trail in the wind they have helped part for you. The effects of this feel greater the faster you and the other person is traveling. The effectiveness of the slipstream is relative to the speed and distance you follow.
Even runners use the slipstream, and they run at a max of 12mph, but I have enjoyed drafting at considerably slower running speeds. It also depends on the strength of the headwind.
And, as Teddy pointed out, the distance you are from the rider in front is important. This is why runners can draft at relatively slower speeds, although for a skilled cyclist (e.g. one that rides rollers all winter) you can ride just about as close as you could run next to someone.
Furthermore, unless you have a direct headwind, you should ride off to the leeward side. For example, if the wind is coming from the right(left), your right(left) hand / shoulder / arm should be as close to the left(right) butt/thigh of the rider in front as you feel comfortable. Put your head down and enjoy. That is the place where you can hang on when the going gets tough.
But remember a golden rule - whoever's front hub is in the lead has the right of way (e.g. if you are drafting, you must yield).