Why are velocity and kinetic energy not proportional?

Question

In the equation $E_k=\frac12mv^2$ the fact that the kinetic energy of a moving object is not proportional to it's velocity seems counter-intuitive to me.

Put another way, it seems intuitive to me that if something with mass has an amount of kinetic energy when it's moving at a velocity, then if you doubled the velocity but kept the mass the same, then the energy would also double.

I'd really appreciate it if someone could supply an intuitive explanation.

Possible duplicate of Why does kinetic energy increase quadratically, not linearly, with speed? — AccidentalFourierTransform, Sep 02 '19 at 20:20
Thanks for that @AccidentalFourierTransform - however with my level of physics-foo, none of those answers are what I would call intuitive. — tom redfern, Sep 02 '19 at 20:37
There is no reason to expect physics to be intuitive. Much of it is anything but. — G. Smith, Sep 02 '19 at 21:39
"it seems intuitive to me ... if you doubled the velocity but kept the mass the same, then the energy would also double." - what is your intuition regarding momentum? — Alfred Centauri, Sep 02 '19 at 22:20
@G.Smith, I agree with your comment regarding the non-intuitive nature of physics. In fact, many first time students of physics have "intuitive" misconceptions in mind regarding physics, especially after seeing the Hollywood version of physics (e.g., Ironman). After teaching high school physics for 13 years, I often found it practically impossible to get students to unlearn these misconceptions, which means that in certain areas of physics, those students were unteachable. — David White, Sep 02 '19 at 23:21
On what of kinetic energy are your intuitions based? Knowing this, we can make progress. — Philip Wood, Sep 03 '19 at 08:55

Why are velocity and kinetic energy not proportional?

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