How many watts is a good push worth? The other day as I commuted home a stranger gave me a helpful push as I approached the top of a steep hill. It was wonderful, and I immediately wondered if this is what an e-bike boost would be like. 

Is there some way to calculate how many equivalent watts a nice push produces? Very curious to know if it's on the low end (20 or so) or the high end (200 or so).
The faster you're going the more power you get out of a given amount of force or "push". Ad bsdl says, power = force times speed, so the same force at a higher speed requires or provides more power.

Going slowly up a hill even quite a strong push doesn't cost much power. A 200W "push" at 10kph will take you up most road hills with little to no pedalling, since that's all the power that's required. A 200W push at 70kph on the flat is noticeable but only just, because 200W requires less force and the rider is already having to put out 500W or so to go that fast. The extra 40% power also has a reduced effect since air resistance goes up as the cube of speed. Adding means 1.4x as much power but the speed change is roughly 1.4^1/3 = 1.11, so you'd only go ~8kph faster or 78kph.

You can experiment with this if you are the one giving the push. The relationship between how fast you're going and how much effort your legs have to put in for a given push force is quite perceptible even at relatively low speeds. It's very easy at a moderate speed to push someone and find that you're going backwards relative to them quite dramatically, but without their speed changing much at all. They'll be saying "go on, push" or "that did nothing!" while you're sprinting frantically to catch back up to them.