Power - why do we use the driving force over the resultant force? I'm an A-Level student, studying Further Maths. We've chosen the Further Mechanics as our optional module of the course.
Further Mechanics, unfortunately, is not making a lot of sense to me, because a lot of it isn't clicking whatsoever. I have two questions which I'm asking here because they also come from the physics A-Level course (I have friends who do it, and in retrospect I probably should have too).
Here are my questions:


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*The equation for power is: power = driving force • velocity. While I understand how this equation is derived, but this is what I don't understand: if the velocity of an object is affected by the forces acting in the direction of its motion and in the opposite direction to its motion, why do we use the driving force, i.e. the force acting in the direction of motion without the resistant forces, as opposed to the resultant force?

*Say we have an object that moves to the right across a rough horizontal surface. The force R is the normal force that acts against the force of gravity. Why is it that, as opposed to using the force of gravity as a source of resistance, we commonly express the resistant force against the direction of motion as μR, where μ represents the coefficient of friction? Surely it would make more sense to use the force of gravity, the thing pushing it further to the ground, than to use the normal force? Of course in many examples they'll be the same, and so it wouldn't make a difference. But why opt for the normal force going up, as opposed to the gravitational force weighing it down?


I do apologise if any of what I've asked is ambiguous or vague, this is a very confusing area of maths for me.
Thank you!


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*RN

 A: 
why do we use the driving force ...

Depends on what you're asking about.  The formula gives you the power delivered by the driving force.  If we think about a car cruising down the highway, I'm not interested in the net power on the car, because at constant speed it must be zero.  I'm interested in the power that the engine requires to overcome friction and drag.

Surely it would make more sense to use the force of gravity [...] than to use the normal force?
Of course in many examples they'll be the same, and so it wouldn't make a difference.

Ah, but when there is a difference, what does it mean?
If the item is just rolling along, your'e correct that they will generally be the same.  But imagine it's a car and you crest a hill at high speed.  The car might lift off the ground temporarily.  The weight (force of gravity) has not changed, but the normal force has gone to zero.
Which of these two is important for friction considerations?  Only the normal force.  If we hit the brakes, nothing happens until it comes down and gets some weight to push the tires down onto the road surface.  We can imagine many other situations where the weight is joined or opposed by another force to change the normal force from the road.  So while in a static situation the weight might be driving the normal force, they are separate and it's not a good idea to substitute one for the other.
