Simple Answer:
With current technology capacitors are not a viable option for powering an electric bike.

Medium Answer:
Currently, the energy density for ultra capacitors is at best about 1/10 that of lead acid batteries so it would take about 10 times the weight in ultra capacitors as it would for lead acid batteries.  Unless you have room for 300lb of capacitors this isn’t really an option.

A little wiki action:

(3–5 W·h/kg for a standard ultracapacitor, although 85 W.h/kg has been achieved in the lab as of 2010 compared to 30-40 W·h/kg for a lead acid battery)

http://www.nesscap.com/product/overview.jsp

http://en.wikipedia.org/wiki/Energy_density

http://en.wikipedia.org/wiki/Ultracapacitor

Long Answer:
Even short trips require 4 or 5 amp hours of power at the rated voltage to be effective.

Others have asked the same question on multiple web forums, and the best answer I could find was this.

1 Farad is 1 Ampere second per Volt. 1 F = 1 As/V.
1 Ah = 3600 As
3600 As / 12V = 300 As/V = 300F
But you need more then that because the voltage will drop as the capacitor discharges. Half the charge means half the voltage.
With a lead acid battery that’s different. A battery that is half empty still has nearly the same voltage as it had at the start.
Maybe 600F and some electronics to automatically step up the voltage so it stays at 12V.
So you’d need to buy 160 capacitors with 3V and 60F each.

http://www.physicsforums.com/showthread.php?t=256751

Even after reading the above statement I decided to do more research and see if there is any new technology that could be useful. Ultra capacitors are starting to come down in price and are readily available, but they still are not cheap.
It is possible buy capacitors that are rated for 3000F and 2.7 volts, and their data sheets state that they have 3.30 Wh (~1.2Ah = 3.3Wh / 2.7V) of usable stored energy. The problem is that the calculation does not factor in the voltage drop that takes place as the capacitor discharges.

Ideally, with a bike you would want to run somewhere between 60 and 36 volts for a hub motor.

There are two options:
1. A voltage multiplier circuit.
2. A string of ultra capacitors in series.

The first idea is probably the better one, but it is fairly technical and goes beyond the scope of a blog post.
The basic math behind a series string of capacitors would be this. (Capacitance drops as capacitors are added in series.) 1/((1/3000)+(1/3000)+(1/3000)+(1/3000)+(1/3000)+(1/3000)+(1/3000)+(1/3000)+(1/3000)+(1/3000)+(1/3000)+(1/3000)+(1/3000)+(1/3000)+(1/3000)+(1/3000)+(1/3000)+(1/3000)) = 166. 7F

Theoretical storage would be
166. 7F*48V = 8000As
8000As/ 3600 As = 2.2 Ah

However, the usable potential is likely 1.2Ah which is stated on the data sheet, and the hub motor will operate most effectively between 48 and 36 volts. That is a 12V change and happens to provide access to about 25% of the usable 1.2 Ah in a simple series circuit.

……………………………. This is getting too boring.

If you still want to know more about sizing ultra capacitors for electric vehicles this article provides the required calculations. http://people.clarkson.edu/~pillayp/c1.pdf

Conclusion:
Start considering ultra capacitors for application in electric bikes  when ratings of 12000F + capacitors become available.  There were a lot of claims in 2007 and 2008 that we would see the end of batteries, but it looks like someone is getting close.

http://physicsworld.com/cws/article/news/44477

For about $16 I grabbed a motion sensor switch that automatically turns off on a variable timer.  I would have been able to install the switch in about 3 minutes if I didn’t have to track down all of the mislabel circuit breakers in my apartment. I’m a little concerned that whoever wired my apartment simply used “lites” four different places in the breaker list.  It doesn’t exactly bestow great confidence in how the rest of the apartment is wired.

Once I had the cover off the original switch box I found some more 1970′s awesomeness. They had simply left the ground wire dangling in the box and jumper in the switch into place with some outlets.  It isn’t anything to be scared of, but the apartment could use a little updating.  Regardless the ground is now being properly utilized, and the switch isn’t just jumpered into place.

We will see how well it works, and if I like it before I do any more.  There really is no point for me to update something I’ll have to undo someday.  Also, we face the problem that most of our switches face the opposite direction of the rooms that they are actually controlling. Whoever wired my apartment had the revolutionary idea of being able to control lights before you actually enter the room.  Motion sensor switches require heat variations in the room to operate, and if they don’t have a direct line of sight they won’t work.

Update:

I moved the switch from my computer room to the bathroom after buying the proper faceplate.

Originally the layout of the switch was something like this.

Light Switch Outlet

Diagram from http://www.indepthinfo.com/

Old Wall Switch

Old Wall Switch Faceplate Removed

Old Wiring

New Wiring Motion-Sensor Switch

Setting up the timer and light sensitivity

Finished Motion Sensor Light Switch

Round Two

Old Double Wall Switch

Stuck Faceplate

If painted over make sure to gently cut around the edges of the switches or you could cause damage the paint prying the switches off the wall.

Old Switch Removed: Discovered that there is no accessable ground wire...

New Motion Sensor Switch Installed

Troubleshooting:

After getting it wired up I turned the circuit breaker back on and found that my CFL bulbs flickered and did not want to stay on.  The bulbs did not flicker in the computer room what is different?  So after a short search I noticed that I had a single incandescent bulb in the fixture in the computer room.  I moved the incandescent bulb into the bathroom sure enough all the lights now function properly.  Like most electronics you get what you pay for.  Apparently the particular motion sensor switch I bought does not do well with certain CFL bulbs (6105M-PT). After reading some forums I found it is best to avoid Cooper Wiring Devices model 6109 and 6105.

The 6119 model should work better with CFL bulbs.

My work around: Currently I’m just using a single 40W incandescent bulb with two CFL bulbs and it works just fine.  It is possible that it is not working because it is not grounded properly, but I really don’t think that is the problem. I might take it back apart and dig around to see if there is a ground wire stuffed in the back somewhere. Even if I find one I do not think it will allow me to run CFL bulbs.

Even though I have to run a 40W bulb it will still save power because the light will not be left on for 8 hours at a time.