What you are describing is common with hub motor ebikes. Worth noting: almost daily I see people in Rad Rovers, Rad Wagons, Rad Runners and Lectric Xpeditions (3 guesses what the common bikes are in this town, especially for the bike rental places we have all over the place) walking their bikes up some of the steeper hills here because of what I am about to describe. <snip>
You assume that they are pushing because the hub motor isn't torquey enough. It could be that the battery is dead or dying. Or they left it in top gear, so they are essentially not helping at all. Flippin' muggles.
Unfortunately, generally speaking that is not true.
The wattage rating of the motor is about how much sustained current input can take. Has nothing to do with output.
Wattage (or power) is only partly about current, since it's defined as Volts x Amps. I'm not sure why you feel that it refers to input power instead of output power. Lacking any documentation, I would assume the opposite, as it's motor OUTPUT that people are interested in. It may need 1,000 W in to produce 750 W out, depending on the motor's efficiency.
What makes a motor go 'round faster is higher voltage from the battery system. As in, going from 36v to 48v to 52v and so on. More volts translates to more motor rpms.
False, at least for brushless motors. The controller and its electronic commutation is what determines RPM. You can have a 24 V brushless motor that spins faster than a 52 V brushless motor if the motor controller is set up that way.
More amps on the controller translates to more power to get that motor up from a start to those rpms.
That's true, assuming the same voltage, since power = volts x amps. (Watt's Law)
A very, very general comparison in automotive terms that are much more broadly understood is voltage = horsepower, and amps = torque.
This is a horrible analogy; you can't really equate these units. You should equate electrical power, in Watts to mechanical power, in Watts, Horsepower or Joules.
What matters is that power denotes how much actual
work can be done. If you have a controller that provides 10 V @ 1 A, its theoretical max output is 10 W. If you have a controller that provides 1 V @ 10 A, its max output is
also 10 W, due to Watt's Law. Either situation, if the motor and controller are set up properly will yield the same mechanical power available to the motor. That said, higher current requires bigger conductors, and copper is expensive and heavy, so higher voltage is the preferred way. (though higher voltage requires better electrical insulation and/or bigger spacings)
Looking back at your analogy, if we see an eBike rated 52 V, we are initially impressed. The educated among us will next look at the ampacity of the controller, since 52 V means nothing if it's choked back by a controller's lower current limit. Then, we look and try to deduce whether the battery pack can supply that amount of current.
Put another way: We can have a 52 V system with a 19.2 A current rating on the controller, and that is 1,000 W available to the motor.
Or, we can have a 48 V system with a 20.8 A current rating on the controller. Here too, 1,000 W is available to the motor. One rating means nothing without the others. In either of those cases, if the battery pack can only supply 15 A, then the 52 V system will be more powerful, since 52 V x 15 A = 780 W and 48 V x 15 A = 720 W.
You're not doing any one any favors by over-simplifying. That's what politicians and executive managers do that really messes things up.
We do appreciate what you're trying to do though.