I'm not looking for how it's derrived. I'm wondering why it works out this way that doubling the speed quadrouples the KE.
Here's my thought process:
A car is moving down the road (ignoring air resistance and any other external force reliant on speed) at 10 m/s. To get to 20 m/s the engine should burn x amount of gas, containing x amount of energy. To do the same acceleration over again it should burn x gas again, resulting in 2x gas burned total and resulting in another 10 m/s increase in speed. This would leave it at 30 m/s.
Say the amount of gas burned to increase speed by 10 m/s is 100 energy units, and the car's mass is 2. Then increasing from 10 to 20 with ke = 1/2 mv2 should increase the KE by 100. Therefore from 20 to 30 it should also be 100, resulting in 200 total energy units and 2x gas burned. But using the equation and plugging in the same mass value with the total acceleration from 10 to 30 (30 -- 10 = 20 and plugging that in for v rather than 10) you get the total ke as 400 units rather than 200, but you theoretically should have still used 2x gas.
If someone could explain to me what the issue is in my reasoning and explain why speed is squared, that would be much appreciated.