a 1 farad capacitor will discharge 1 amp for 1 second so a 1/8 will do 1/8 of that, and you will need to put several 15v caps in series to get enough voltage, so that will cut the capacitance by the number in series. let's say you put 4 in series to stand off 60v, the least you would want to use on a 48v system, you would end up with 33,000 uf. this is enough to discharge 10 amps for about .003 seconds. you will not see any significant reduction in voltage drop when you hit the throttle, except for the first few milliseconds. the caps just cannot store enough energy to run the motor for any significant length of time.
the caps in the controller are necessary to maintain a stable dc voltage for the fets which switch very quickly, and they are kept charged by a low impedance path to the battery. a precharge resistor will cut down on the inrush of current that occurs when you connect the battery, and will mostly eliminate the spark you see, which will save wear and tear on a power switch if you have one in the battery wire. in the controllers most of us use the caps are kept charged all the time and the switch controls a transistor circuit that provides power to the controller logic circuits.
supercapacitors that have 10 farads or so would provide some real output current, but only for seconds at a time. they are just not a good substitute for a better battery.
in a car stereo the big capacitor helps provide the peak current for transients that require much more than the base line power, so the headlights will not dim on the bass notes, but these peak demands last much less than a second. most people these days use at least a 1 farad 20v cap, and one of those caps is bigger than a beer can. even a sixpack of them would make little difference in performance on a typical ebike.
the big caps on a car stereo power line also serve to protect the battery from the high peak current, which can shorten battery life by eating chunks out of the plates. the cap lets the current come out of the battery more slowly so the chemical reaction can be maintained at a safer level
I do not approve of this group any longer and regret what it has become. I could once share ideas here freely that is no longer possible.