Don't ever ask me to calculate anything. But my real world commute has a hill very much similar to yours but a bit shorter. I go about one mile on the steep part, followed by about a half mile slightly less steep. I run a WE brushed hub, at 36 volts with a lifpo4 20 ah battery. It will do 25 mph or more on the flat and about 12 mph up the hill. As I start the hill, which does not get gradually steeper, I can go close to 20 mph for a few hundred feet but then for whatever reason at that point I slow down to about 15 mph for most of the hill. In the last 1/4 mile of the steep part, which may be steepest, I will slow down to around 8 to 10 mph. By then I may not be adding many watts by peadaling though I try. On the less steep part, I will be able to go about 15mph again. When I was using lead, it was much slower, especially at the top of the steep part. 48 volts lead did help, but it still was slow due to voltage sag.
48 volt lithium will get you up the hill very nice, and not so tired at the top of it. If you get a 20ah sise the cells will be able to handle more drain since the strain is spread out among more cells. Big hills can trip the bms of small lithium packs, and the extra range is worth it's weight in gold. If you are on a tight budget, Brushed is pretty fast, and climbs hills better than brushless motors in the same price range. The really fast brushless motors are expensive but well worh it if you have the cash.
I don't see you climbing that hill at 48 volts much faster than 15-20 mph. If you do that at 30mph I'll want to copy your bike for sure!
My brushed motor gets real hot by the time I top the hill, on a 90 degree farenheight day, but nothing is frying yet. It's hot but you can still touch it. brushed seems to be ok at 48 volts from what others say. You mention getting a geared motor, from what I've read they work great at hills and slow speed, but may wear out fast at high speeds. I have no Idea how fast the geared motors are. For me, the brushed hub pulls that hill plenty good enough since I'm not getting off to walk. The majority of my ride is flatter and at 20-25 mph, or faster on the downhill ride to work. So though I do have bitch of a hill to climb, most of my ride is at above 20 mph. The brushed hub is a pretty good compromise between go fast, and pull hills good. If they wear out in 2 or 3 years, they are cheap enough to replace. But if I have money then, I'll be shopping for something that goes 35mph.
My spreadsheet got corrupted, somehow...
Even at 36V's it climbs great though.
