I have recently bought an e-bike, and it looks exactly like a normal bicycle unless you notice the unusually thick strut of the frame, and the (tiny) charging port.
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As Zooterkin says, the limitations here are that the motor must cut out if you stop pedalling, and when you reach about 15 mph. There is another restriction, that the motor must have no more than 250 watts of power. I have read opinions that this last is discriminatory, because it discriminates against older, weaker and heavier riders by limiting the gradient they are able to climb. This opinion says that the maximum speed cut-out should be enough to ensure safety, in that riders are not being electrically assisted to go faster than an average rider might normally achieve, and that more power should be available for people who need it to get up the hills. I can see the logic of this, as I'm in the older and weaker class of rider. I have not yet met a hill that I was unable to climb on the bike, but having said that the steep sections were quite short and I was getting quite out of breath as I got to the point where the gradient lessened again. This was all on the road and things might be rather less favourable on gravel or indeed grass.
All in all though, I love the bike to bits. I cycled all the way up to the Baddinsgill reservoir, where that picture was taken, without dismounting, including one section steep enough that when I saw it coming I thought, no way in hell. Hopefully as spring advances I'll get a bit fitter and the "no way in hell" sections will become more routine. The great thing is that it's getting me out there on the bike. The roads round here are undulating enough to make any ride on a normal bike a bit of a chore on some sections, and catch me trying it in January! But when I know that I'll be able to make the hills without too much exertion and without getting off and pushing, I'll do it.
This bike is actually lighter than my previous bike, and better geared, and I've been running around the village on it without bothering to switch it on and finding it less effort than its predecessor (currently sulking in the garage). Light enough to lift into the car, with the front wheel off, so the world is my oyster. I can even recharge it from the car - the car isn't even going to notice 250 watt-hours, which is the bike battery's capacity.
Regarding delivery drivers in the city, I've seen them. I'm not convinced their bikes are illegal. These are mainly strong young men and the startling thing isn't the speed they're doing but the acceleration. A 250-watt motor will do that for someone who is already fit.