Motorized Wheel Chair Source

Ultra caps have improved dramatically in the last few years. However, each cap is good only for a volt or two, so you need several to get 12V, then you have to parallel them to get reasonable current. Also, as the cap discharges, its voltage drops down to zero. This means you need sophisticated electronics to get much energy out, as motors pretty much need constant voltage to work well. So, a 12V motor won't do much on say 5V.

The weight of the ultracaps is something like 10x to 100x more than lead-acid batteries. The cost is something like 100x to 1000x the cost of the batteries. I figured these out once, you can do it by googling ultracaps on the internet to get weight, size, capacity, and cost of ultracaps. Ultracaps are adequate to store energy from braking to a stop and for a quick burst of power to accelerate up to high speed quickly, but this costs something like 2x to 5x the cost of the entire battery pack. They are nice for performance, but just too expensive right now.

For batteries, weight and cost are both important. Best lead-acid by weight seem to be 12V-9Ahr (5 lb weight) and 12V-21Ahr (10 lb weight). For useful power, use 0.6C/hr for the 1 hr rate, so the larger battery will put out 0.6*21= 12.6A for 1 hr. Using an 80% efficient motor, assuming 12V, and using 746W/HP electrical conversion, this battery is good for about 1/6 HP for 1 hr, or 1HP for about 5 to 7 minutes. Best prices for the smaller battery is about $20, best for larger is $30. To get these prices, you likely want to buy cheap surplus Chinese made batteries, which may have less capacity and/or more weight. Fancy batteries have lighter weights, but way more costly.

A bicycle uses perhaps 1/4 HP when cruising at say 15mph, 1HP at top speed of say 25mph. A 50cc moped puts out 5HP at speed of say 35mph.

So, in designing your motorbike, you need to minimize power needed and maximize power output.

A very efficient motor/drivechain is key. One with a 90% efficiency instead of 45% gets twice the performance from the same batteries. Very difficult for hobbyist to get efficient motors at reasonable cost, I have not been able to find. There are a few in-hub motors for front wheel of bicycle that make about 1/3 HP on 12V, and 1HP on 36V. They are direct drive, look like about 65% efficient. They may be your best bet.

Drag needs to be minimal. You want very low friction bearing, like a bicycle or equivalent. Narrow and high pressure tires reduce rolling resistance. Some streamlining will increase both speed and range.

Good luck!