Motor/battery combo

Maybe. Depends on the controller used.
https://endless-sphere.com/forums/viewtopic.php?f=2&t=74403
 
It depends on the battery. Not all 36v10Ah batteries are created equal. Some are only suitable for powering small motors at low amp draws, while others could be used to drag a bus up a pole. What matters is the battery's output current, which is often given as the battery's Max continuous amperage, or it's "C" rating. Many of the 500w kits come with either a 20 amp, 22 amp, or 25 amp controller, so you want a battery that can handle at least that amount. So a 25 amp continuous rating, or 2.5C battery would be able to handle any of those.

Some of the cheaper 36v10ah batteries are only good for 10 to 15 amps of output used for small 250w motors, and would be ruined trying to push a bigger motor at higher amps.
 
Many cheaper batteries are made of lower quality cells that are incapable of pushing out too many amps, Which is your real problem here. If you're getting an Ebay pack I wouldn't count on more than 1.5C. C rating is for Capacity; a 1C pack will put out its Ah capacity, a 2C pack will put out twice its Ah and so forth. (36v*10ah)*1.5=540W of power. So it *will* work. However there's all the itsy bitsy things like pack internal resistance, life over cycles, cycle life, imbalance etc. Here's a quick rundown:
1) A high quality pack has low resistance between the batteries and as a whole
2) It can provide at least 2C-Just like a car, having more than you need makes it last longer
3) Draw less power and charge slowly up to 4V per cell, you can get upwards of five hundred cycles.
4) A pack you want should also fit in your frame or rack well. In frame is ideal because the bike shields it from drops and it doesn't move your center of mass
5) Have a good BMS or smart charger- A BMS makes sure the batteries are evenly charged during use (as well as some other features) and a smart charger during recharge
-Note: That's why BMS is so popular with shitty packs that tend to get out of balance constantly-Sometimes 0.03V over half a cycle!
6) The manufacturer is important. 'Ultrafire' isn't a battery brand, it's a brand of shrinkwrap! Lots of 'ultrafire' style batteries are literally factory rejects; it's cheaper to sell them than to pay to get rejects disposed of. I despise this style. Best you can get (BEST) are panasonics, second being LG and samsung.

That's as simple as I can get it. If you need help picking out a specific pack, I'd be happy to help you
 
It should be ok, but I can only guess in a general way at it.

find out what amps the controller is. Likely 20 or 22 amps. Then match that with a battery that lists its max amps at 30 at least. In most cases, even the cheaper cells out there these days can handle 30 amps, in a 10 ah size battery. Then, with 30 amps possible, and 22 your max, you should be fine for the battery to last quite a while. However, a battery listing max amps as 40, with your controller only 22, should have an easier life, and perform better for longer. Good cells can be worth their cost. Or go bigger with average cells, and get the same 40 amps max.

If your range will be a bit short on a 10 ah battery, ( figure on one mile per ah riding fast) then get a larger battery. Bear in mind, just because you have enough range when the battery is new, doesn't mean you will by next summer. So if your ride is 8 or 9 miles, you'd eventually need to slow down some to make it. Figure on 2 miles per ah if you only ride 15 mph. IMO, the real reason to go big is to increase your ability to go far on the bike, and leave that car parked more. I tend to never leave the house carrying less than 20 ah, and often carry 33 ah.
 
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