Why does kinetic energy increase quadratically, not linearly, with speed?

That’s because kinetic energy is the integral of force operating over a distance. If we apply a force of 1 Newton to a mass of 1 Kg at rest for 1 second, the the body will accelerate at a uniform rate from 0 to 1 m/s and, in doing so, will have covered 0.5 metres. That means the energy expended will be 1 x 0.5 = 0.5 Joules (a Joule has units of Newton-metres).

Now say we apply another force of 1 Newton to that 1 Kg mass now moving at 1 m/s for 1 second. It’s going to accelerate to 2 m/s. However, unlike in that first second, the object is already moving at 1 m/s, so the total distance it will have moved in this time will have been 1.5 metres. Thus, this time, the energy required was 1 x 1.5 = 1.5 Joules. With the energy expended in the first interval, that is now 2 Joules. So, getting the object to 1 m/s required 0.5 J whilst doubling it to 2 m/s required a total of 2 J, of 4 times as much for doubling the speed.

Do this again for a third second, and you get 1 x 2.5 = 2.5 J, and then for a fourth second, it’s 1 x 3.5 = 3.5 J at 4 m/s. Thus out body at 4 m/s has 0.5 + 1.5 + 2.5 + 3.5 = 8 J of energy, or 16 times as much as it had at 1 m/s.

I’ve deliberately made this as simple as possible using basic numbers. It’s not meant to be a formal proof as it tackles a special, and not the general case. It’s there to illustrated the principle.

For the general case it would be necessary to allow for variable forces and use a bit of calculus, but it would ultimately come out with the formula  Ke=12mv2Ke=1/2mv^2 .