Let’s chose as reference the a point on a road. And let’s accelerate a car. I assume you know the equations. Let them assume we use the car’s engine to generate the force, and this engine is a combustion one, and we never shift gear, fix gear ratio.

We start the reasoning with: Speed = time * force / mass.

It happens that the car will travel faster and faster while accelerating, covering more and more distance per second. Distance=time squared * force / mass.

I will keep using force/mass as a substitute for acceleration.

Now, let’s say you push ed the car with 1000force, for 1000 distance, it will get to 10speed. To get the same car to the speed of 20, it will take twice the time.

But how much distance it took? To get to 20 speed instead of 10 speed, we will need twice the time. Again, distance=time squared*force/mass. So we will cover 4 times more distance as 2time squared is gives 4time as a result.

Breaking it down to the combustion in the engine, while the car gets faster and faster, the engine spins faster and faster, burning fuel faster and faster. There will be a fix ratio between engine and wheel, for each revolution of the wheel there will be a fix number of combustion cycles, all burning the same fuel. Pause it at any point of the travel: you can see for each meter of road, you burn a fix amount of fuel.

If getting to twice the speed covers four time more road, therefore burning four time more fuel.

This means the car has received 4 times more energy to get to 20speed compared to the energy it takes to get to 10speed.

From energy pov, the time it took doesn’t matter, what matters is what you burn to get there and for how much distance you had to burn.

The car’s kinetic energy is nothing else that the burned fuel energy. And the amount is determined by how much force for how many meters It traveled under that force.

To recap: kinetic energy=“fuel used” to get the object to that speed.

(speed=force * time /mass, transform this you have:

force= speed * mass / time.

Distance= speed * time)

SO HERE WE ARE: Kinetic energy = “fuel used” = force * distance = (speed * mass / time) * (speed * time) = speed * speed * mass / time / time= mass * speed squared

Hope this example helped visualize it. The rest of your question is answered just by changing the reference point. Law applies the same.