Well, you got us to look. Brains are sexy too.
But back to the subject at hand. Under 5 miles, even the crappiest ebikes work fine, but hills are another matter. The site I like for finding out how steep a hill is is calle Map My Ride. Once you know how steep, you can decide if you need a gearmotor, or a monster direct drive or what.
My rule of thumb is , under 5% hill grades, all the 500 watt and above direct drive motors work ok at 36v. Direct drive is the kind that is nearly totally silent, and is happiest when run about 15 mph or faster. A 9 continent motor, sold as Ebikekit by one good vendor, handles 7% just fine with 36v. This motor is a bit larger diameter, and the extra leverage helps it climb.
Another approach is the gearmotor. This type is usually lighter and smaller, and better at traveling slower up steep grades without overheating . The motor inside spins faster, and a planetary gear inside results in good pull at normal speeds. These motors also feature a freewheel inside that makes pedaling easier if the motor is not used. Often lower wattage than dd motors, they climb great, but possibly a bit slower than a dd motor that has more power avaliable.
Look at Ebikes-ca and EbikeKit websites for motors, both excellent vendors you can trust with great customer service. You need to choose the motor first, and then based on the motor, controller, and the kind of riding distance and grades, choose the battery.
The battery will need to be big enough to power the motor up hills without killing itself. You will be maxing the power climbing hills constantly, but range will not be a problem since a powerfull enough battery will have 10-25 miles of range. Lithium is the best, but often it is sold in a 10 amp hour size for bike kits. While quite adequate for normal riding, I like to advise a larger size battery for steep hills. Some can provide enough amps in 10 ah size, and some can't. I'd be looking at the battery specifications, and find one that puts out 30 amps continuous power or more. Many of the lifepo4 batteries in 10 ah size put out only 20 amps continuous. Some other lithium chemistries can do better LiMn, or better quality batteries like A123 lifepo4, pack a punch in a small package. Mostly those are found in drill batteries, and several can be strung together to get big power in a small package. That way you don't need to carry a big battery box.
The approach I used myself, was to simply buy a bigger size lifepo4 pack. That suited me best since I ride 15 miles one way. Pingbattery is a good source for that. 48v 15 ah powering my ebikekit motor gets me up 7% grades at 18 mph. The whole thing costs about $1000, motor and battery.
