Thank you all, for trying to help me “get” it. I understand the very straightforward definitions of “work” and “force”, and Pasta got me to a breakthough by pointing out that kinetic energy also involves velocity squared, which is precisely the idea that I’m trying to wrap my brain around.
So I found this in the Wikipedia article on Kinetic Energy:
By dropping weights from different heights into a block of clay, Willem 's Gravesande determined that their penetration depth was proportional to the square of their impact speed.
Aha! So if I drop two equal weights, and one has triple the speed at impact, it will penetrate nine times as far? Okay, now I have to digest this a bit… Thanks!
Depends. For instance, see the description of Newton’s approximation of the impact depth of a blunt, high-velocity penetrator: as long as it is going fast enough to plough through the target, the penetration distance will not depend on the velocity at all.
This is not correct. And does a lot of explaining that involves more not less hand-waving (like dt is 1 second !?) that gets us further from the physics.
As mentioned above, that webpage is unhelpful. It makes some sense if you already understand the underlying concepts but can easily be interpreted poorly (as demonstrated in this thread already). If a “simple” explanation is inexplicable unless one already understand the “complex” explanation, it is of little value.
Properly speaking, Force is the first derivative of momentum with respect to time. F = dp/dt. That’s it. No need to make it any more complex than that.
That’s separate from the equation you presented. In the case of Newtonian mechanics, Force does simplify to F=ma. Where we go wrong is in the case we’re not dealing with Newtonian mechanics. When relativity comes into play, momentum is no longer observed to be ma but can (after some algebra) be expressed as ma plus some “other stuff”.
But F=m*a no longer being strictly true flies in the face of a lot of education, so in the past that “other stuff” got wrapped up into a “relativistic mass”. But that’s not very useful in any real way. Better not to get involved with it at all.
The derivation assumes that in the limit of v goes to c that any energy added to a particle does not change v. Taking the equation for the energy of a variable-mass system, it assumes therefore that any energy must change the mass in the limit of goes c and shows this change is E=mc^2. The issues with the derivation are firstly that the version presented is dumbed-down to a level it loses any explanatory power, it assumes off the bat that accleration produces a change in mass and it only shows what happens in the limit of v goes to c.