this was a very nice question and i actually enjoyed answering it very much! it is a conceptual and experimental derivation (or confirmation) of Newton's Second Law. i think the asker really enjoyed my response and gained a good base for understanding the law.
Q:
If work is: force x distance... as well as: mass x acceleration, how can I relate these two ideas and think of?
force? Okay - let's break it down - we multiply mass by acceleration to get a force, which makes sense. I can think of it as an acceleration per mass, which takes force (and hence energy supplied). But if I have W = force x distance or W = Fd, I can equivalently say: F = W/d. So force equals work done per unit of distance. Which makes sense to me if I think of the fact that the two equations I have pointed out here require energy, that is again: mg and W/d.
Am I over analyzing this too much, or how can I really understand force? What throws me off is that I have two different equations for force - one is a product of two variables and the other is a quotient of two variables. Why couldn't I say F = m/g?? Or also say: F = Wd????
Additional Details
or instead of F = m/a say:
F = a/m
A:
a good way to think about these things and really get a good conceptual idea for what these equations mean is to create examples for yourself. sure, they wont get you the exact form of the equation, but they'll give you a rough idea of what needs needs to go where (like why mass is *multiplied* by acceleration to get force).
so first, think of newton's equation F=ma. what this says is that in order to move an object of mass 'm' with an acceleration 'a', you need a force 'F' equal to their product. so how does this make sense? well it'll be easy once you think of a few examples that you run into in everyday life!
imagine you are trying to push (or pull) a box along the ground. in your experience, does it matter if the box is REALLY heavy? of course! if the box weighs 2000 pounds, there's no way you're going to be able to move (or accelerate) it! you would need a much larger force to push it along! so then in our formulation of newton's law, we know that if you *increase* the mass of the object, you should also have to *increase* the force it takes to move it, right? well that makes sense, and it seems kind of obvious now! because it is certainly easier to move a feather than it is to move car - and the car even has wheels! so where should we have to put the 'm' if we want 'F' to increase as 'm' increases? well, certainly not in the denominator! if we did that, the force would *decrease* as mass increased! now that would be a sight! so we know 'm' has to be somewhere in the numerator!
now onto acceleration. again, let's use something we're familiar with! let's look at that box again. let's say you want to get it going up to 1/12 of a foot per second (an inch per second) in 1 minute. so you have all of 60 seconds to accelerate this box to a final speed of an inch per second. do you really think it takes a huge amount of force to move something that slow (about 0.0014 ft/s² or 23000 times slower than the acceleration of gravity!)? no, your baby sister could easily do that! but what if someone told you that they need to get this box going 100 miles per hour in the span of 1 second. could you do it? is that even possible?! (assuming the box isn't moving when you start, that would mean the acceleration would be 147 ft/s², or about 4.6 times greater than the acceleration of gravity!). well yes it can be done, but could *you* do it? it's nothing personal, but i don't really think you - or anyone else for that matter! - is that strong. it would require some very massive force to do it! (keep in mind the best cars can go from 0-60 mph in about 3 or 4 seconds, which is pitiful compared to what i'm asking you to do! and they've huge 500hp motors to do it!). so the conclusion is that the bigger acceleration you want to give to the box, the bigger the force you have to impart on it will need to be! and we know - from the previous example - that this requires 'a' to be in the numerator! therefore, F=ma, and there's no other way it can go!
so there you go! this was a rather simple method, but the power of this analysis is incredible! see if you can apply the same thinking to your work/energy problem and get the correct answer!
remember, all that physics is trying to do is describe the world around you. so all you have to do in order to understand it is play around with it! it's right at your fingertips :]
so try this same method with the other problem and i think you'll be pleasantly surprised at how physics describes the world. and if you have any other questions, i'd love to help you out some more! :]
happy hunting!
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