A friend of mine had an idea to attach a motor to a bicycle with the idea to be able to turn it on and have it maintain average bike speed, apparently 15-20 mph. Neither of us have taken physics classes before, though, so we aren't too sure how to calculate how much force (torque?) the motor has to put out. Any help would be appreciated, and I apologize for my general ignorance (and Imperial units).
It would be difficult to calculate the power required since there are so many variables. faced with a problem like this your average physicist (or more likely engineer) would try a couple of different motors and see how fast the bike went. Once you have some data you can then use a mathematical model to optimise the design.
Actually, what your average physicist (or especially engineer) would actually do is a literature search, known these days as a Google, to see if anyone else had already solved the problem. In this case he'd rapidly find http://www.powacycle.com/ and discover that the motor power needs to be about 200W.
All you want is a ball-park estimate, right? So ride your bike on a level surface in typical wind at typical speed in a typical gear. Estimate (roughly) how much downward force you are exerting on the forward pedal. Multiply that by the crank radius. That's the torque you are applying to the crank.
Now multiply by the number of teeth on the rear chain sprocket, and divide by the number of teeth on the front sprocket. That's the torque at the rear wheel.
Sometimes people get hung up trying to get needless precision, when the problem isn't very precise anyway, and a little common sense gets it done.
Do you want to bike uphill with a slope of 75 degrees, while carrying a person of 450kg? Do you want it to work in hurricanes, underwater, or while dragging a caravan behind it?
Before you dive into the numbers, it is a good idea to write down a clearer definition of your problem. Only then can we begin tackling the physics behind it :)
Having said that, why do you want to know this in the first place? It is far easier to have a feedback loop, e.g., a system that measures the bike's current speed, which increases or decreases the power applied to the breaks and pedal-assistance accordingly. At this stage, start thinking about safety too -- what would manual breaking accomplish if the system will force your bike to keep going at 15mph?
The maximum power will basically only affect the maximum load under which the motor can maintain the 15-20mph speed requirement.
I foresee different product lines, where a cheap 100W motor serves a city-dweller in a hill-less country, and an expensive 5000W motor serves a mountainbiker with heart problems.