How much force is required to propel a bicycle? 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).
 A: 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.
A: 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.
A: More important than the "force" - which is what you are asking about - is the "power" of the motor (whether electrical or combustion). As a rule of thumb, most people can easily produce 150 W of power; you work yourself into a serious sweat at 250 W; and professional cyclists in a sprint may get up above 1000 W (for more details see http://www.bicycling.com/training-nutrition/training-fitness/guide-power-meter-metrics). An interesting metric from the same site (different page) is "power per kg" - with guidelines of 2.5 W/kg for an untrained person, up to 6 W/kg for Tour de France contenders (for sustained riding).
Assuming you have a bike with low friction (thin tires, well inflated) on a flat surface, then 200 - 250 W of net power will be enough to maintain a speed of 20 mph.  If you use a 12 V battery, this means about 20 A; and since the capacity of batteries is often expressed in "Amp-hours" you can see how a 20 Ah 12 V battery might last for an hour (in practice you never get all that, because of losses in the battery, the motor, and transmission). Add starting and stopping, and hills, and the power requirement goes up significantly (you might want at least 4x that for peak power).
Now for the second part of your question: "how much force"? If you have a 25 cm diameter front sprocket, and you are pedaling at a cadence of 80 rpm, the chain moves 80 x 25 x pi cm per minute, or about 1 m/s . 20 mph ~ 9 m/s . In other words, the mechanical advantage of the chain driving the little rear sprocket is about 9x, which means that the force on the chain is nine times greater than the force you would experience if you were holding a piece of string whilst sitting on your bike and someone was pulling you at 20 mph.
1 m/s and 200 W means a force of 200 N on the chain. If you want to connect a motorized drive wheel, you will need to be able to generate that much force on the edge of the drive wheel. If you have a 5 cm radius wheel you would have a torque of 10 Nm (0.05 x 200), with a speed of 400 rpm (5 times greater than the speed of the pedals).  Make the drive wheel smaller, and your torque requirement goes down whilst the rpm goes up.
All the above are ball park numbers. To confirm, I did a bit of searching online, and found a wheel with integrated 500 W motor that claims "high 80% efficiency", requires 36 V battery (higher voltage and lower current improves the efficiency - less heating of the wires), and speeds of 25 kph (about 15 mph).  http://www.thelashop.com/36v-500w-26-inch-front-wheel-electric-bicycle-motor-conversion-kit.html
A: That depends...
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.
A: In the last few years computation of output power has become quite common between cyclists, and tools for fast estimates have spread out, for instance this.
You will find that an average trained cyclist is able to output 400 W continuously, while the sprinters can go up to 2 kW for a short time (hitting ~45 mph). Clearly by adding the weight of a motor and a battery pack you will need more to get the same performances.
