Physics Question Noodle Scratcher

[Kyle85]

OK so I'm toying with a physics question in my head related to arrow speed and force.  Maybe I'm thinking about this wrong and this is a moot point, but here goes:

Adding mass to a limb tip will slow the limb and thus slow the speed of the arrow (as far as I understand and believe).

However, force derived from an impulse measured in Joules (J) is a function of the force applied (F) and the time it's applied (T). So J=F*T which is why a bow that is 40#@26" shoots slower than a bow that is 40#@30" because the longer draw increases the time (T) in the formula that the force is applied, resulting in a faster arrow even though the draw weight (F) is the same. 

So... could you not get a similar increase in time (T) by slowing the limb down with added mass in the tips, even though this does not change the draw weight (F)?  Wouldn't that add force (J) to the arrow?   

This doesn't jive with measuring slower arrows that result from slower limb I don't think - unless the energy is captured somewhere else?  Or does the slower limb reduce the power after all?  Or is the slower arrow irrelevant because this is something that adds momentum at the cost of speed? 

My own noodle's explanation for why added mass would decrease power: does the added mass and slower limb delay how long it takes the F to come up to full power, thus making the cumulative balance of F*T for every inch the string drops lower than if a light limb reached full power quicker?  Does the added mass maybe prevent it from actually reaching as high a value of F despite being an equal draw weight - because the force is absorbed by the inertia of the heavier limb instead of the arrow?  Perhaps that's the answer but what do you think?

If I'm completely crazy and wrong please feel free to criticize me for giving you an unnecessary headache. 

[Orion]

"Does the added mass maybe prevent it from actually reaching as high a value of F despite being an equal draw weight - because the force is absorbed by the inertia of the heavier limb instead of the arrow?" Yep.  Another way of saying its is, slowing the limbs reduces their efficiency  Long story short, it doesn't work.   

[Kyle85]

Lol, you're probably right - I'm sure I'm over-thinking it.  :-)

[McDave]

I sometimes like to see what happens at the extreme ends of a hypothetical situation.  Imagine that your bow were so powerful that on release, the limb tips were moving at close to the speed of light.  In almost no time at all, they would return to brace height, and your arrow would be launched at nearly the speed of light.  The short amount of time that would be required is not a limitation of the high initial velocity that would be attained.