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ELI5 mv^2/2Imagine a car is moving at a speed of 10 m/s. The driver hits the brakes. How much distance does it need to stop?
The main idea is that brakes have a constant force, and the change in speed is always constant. Let's say they reduce the speed in 1 m/s each second.
The first second the car travels 10 m and the new speed is 9 m/s.
The second second the car travels 9 m and the new speed is 8 m/s.
The third second the car travels 8 m and the new speed is 7 m/s.
...
The ninth second the car travels 2 m and the new speed is 1 m/s.
The tenth second the car travels 1 m and the new speed is 0 m/s.
So the total distance until it stops is 10+9+8+7+6+5+4+3+2+1. If you make a pile of bloks and you put first 10, then 9 over them, then 8 over them, .... you get a nice triange. The base is 10 and the heigh is 10, so the total number of blocks is 10*10/2. [1] [2]
So the car needs 10*10/2 m to stop. You can repeat the calculation with other initial speeds, and the result is V*V/2.
I'm not sure if it's intuitive, but the energy is proportional to the distance to stop.
Another posibility is to throw a toy car verticaly, and calculate how hight it goes. It's the same calculation. The maximal height is V*V/2. I think it's easier to imagine that energy is proportional to maximal height.
[1] If you actually count them, the result is 55 insted of 50, more details in https://en.wikipedia.org/wiki/Gauss_sum , but 10*10/2 is a good aproximation.
[2] If you split the time in half seconds and use smaller and smaller blocks, and then use calculus, you get 10*10/2.