I'd call it Impulse or Work Done or whatever else I actually meant. Your problem is that you haven't really got beyond the stage of deciding what you really want to know or what you mean. We have all told you answers that could apply for a range of things that you could mean. (Please please don't give us another story to illustrate what you mean - I think I shall run out screaming at another mention of Rep Rate)
You start of talking about a force then you suddenly jump to "overall or total force". That's where the nonsense creeps in. Force is Force and it means just one thing. If the man pushed the car for 30 minutes, then I am assuming this would be at a steady speed on a flat road, just against friction??? What you could say is that the WORK he did was Force times the distance the car traveled.
If he pushed with less force, then the car would not be going as fast and it would cover less distance. The WORK done would be the new force times the new distance.
You seem to have some objection to describing things that way. Why? It's the way that the rest of us talk because it fits in with the rest of Physics.
You would be doing less WORK but your description, involving the word 'force' makes no sense.
I have done that for you.
Here you go again with 'total speed'. What you seem to mean here is that the DISTANCE is 1000
(100m/s for 10s). Speed times time = distance, doesn't it? So "total speed = distance" for you? Not good enough. You want to join the Physics club and get answers. Club rules apply, I'm afraid and you have to use the right terms or no one will understand what you're on about.
No - they did more WORK and possibly in less time. This means more Power. The details of what forces were involved are not included in their statement.
Speed is a different thing, is it? But Maths applies to everything. The word 'total' means adding things together, whether it's cabbages or Joules. Some things just can't be 'added together' and adding speeds-at-different-times is as much forbidden as adding forces-at-different-times. This isn't poetry or stream of consciousness - it's strict and rigorous stuff.
Sorry, the last bit just reads like ramblings. I can't see what you're getting at except to tell me a story in non-technical terms. Why does a horse and cart have to be different from a man and a car? The man and car scenario says it all.
You are getting some damned good value out of all this you know.