Thank you and GG for your explanations. Sadly I have to confess I never realised that it became harder to accelerate as you went faster. I think that I imagined that at any speed if you applied a bit of "F" to your "m" then away you went at "a" - it was not dependent on your speed. (and I suppose, if you use the right frame of reference this is actually true).
F = m*a still applies. The difference is the work/energy required to apply that F is dependent upon speed. Newton's 2nd law is actually F = d(mv)/dt. Force equals the time derivative of momentum (m*v), which for a constant mass, m, turns into F=m*a. Once you see that all the relations between force, motion, momentum and energy then I think things will become more intuitive. Here's another relation that I use to convince myself that KE = 1/2*m*v^2:
F = m*a = m*(dv/dt) = m*(dx/dt*dv/dx) = m*(v*dv/dx) [This is called the Chain Rule in calculus, so now we have acceleration described in terms of distance rather than time]
Work = integral(F*dx) [Work is Force times distance, though more correctly, it is the summation of Force times (dot-product) infinitesimally small distances (dx)]
Work = integral(m*(v*dv/dx)*dx) [Now the dx's will cancel and we are left with an integral wrt dv]
Work = m*integral(v*dv) [now simply solve the integral]
Work = 1/2*m*v^2
So this tells me that kinetic energy can be thought of as a moving body's 'ability to do work'.
IMO for all great physicists, the mathematics come first. There's a phenomenon that they cannot
explain with intuition and they use math to describe it. The great ones are then able to break the complicated math down in a way that can be understood by those you haven't or can't perform the math. Without math, there would be no intuitive reason for the Universe to not rotate around the Earth.
As I tend to rely very much on "monkey intuition" (or should that be "canine intuition") I like to always try and relate the world I see around me to these basic properties of physics. For example the "friction toy" - (or a wheeled toy that stores its energy for motion in a flywheel). You push the toy along - it stores internal energy in its flywheel - it then exhibits a an apparent momentum that far exceeds its actual mass and speed. Can this be related somehow to the related properties of momentum and KE? Could the internal rotation of particles be the storage for KE that is then manifested as momentum.
If I'm thinking of this toy correctly it looks like a vacuum , but you can think of mass/inertia as a kinetic energy storing element. As this toy is pushed along the flywheel is accelerated (storing KE) and the toy is probably difficult to push (requires more work than a similar toy with no flywheel). As the flywheel decelerates, it is releasing kinetic energy back into the toy and the toy continues in motion even without an externally applied force (until friction stops it). If I'm thinking of the wrong toy, then please disregard my post.