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How can the kinetic energy equation be intuitively understood?

Momentum is proportional to an object's velocity, and kinetic energy is proportional to the square of its velocity ((mv**2)/2). It's pretty intuitive that if object B is going twice as fast as object A and they have the same mass, B has twice the momentum. It's harder to grasp why it has four times as much kinetic energy. The math is clear enough, but for most of us it doesn't give a feel for why, for example, accelerating a car to 40 MPH gives it four times as much energy of motion as accelerating it to 20 MPH.
Looking at braking seems to help. If you're going twice as fast when you start to brake and your speed linearly goes to zero, it will take you twice as long at constant deceleration to decelerate to zero, **and** the average speed while braking will be twice as great (half your initial speed). This implies that the braking distance is proportional to the square of your initial speed, so you're applying a braking force for four times the distance. Similarly, accelerating a car to 40 MPH will take 4 times as much distance as accelerating to 20 MPH. But that doesn't quite get me to energy.
I think there's just one more step to making the relationship to kinetic energy obvious, but I don't have it.

Momentum is proportional to an object's velocity, and kinetic energy is proportional to the square of its velocity $\dfrac{mv^2}{2}$. It's pretty intuitive that if object B is going twice as fast as object A and they have the same mass, B has twice the momentum. It's harder to grasp why it has four times as much kinetic energy. The math is clear enough, but for most of us it doesn't give a feel for why, for example, accelerating a car to 40 MPH gives it four times as much energy of motion as accelerating it to 20 MPH.
Looking at braking seems to help. If you're going twice as fast when you start to brake and your speed linearly goes to zero, it will take you twice as long at constant deceleration to decelerate to zero, **and** the average speed while braking will be twice as great (half your initial speed). This implies that the braking distance is proportional to the square of your initial speed, so you're applying a braking force for four times the distance. Similarly, accelerating a car to 40 MPH will take 4 times as much distance as accelerating to 20 MPH. But that doesn't quite get me to energy.
I think there's just one more step to making the relationship to kinetic energy obvious, but I don't have it.

classical-mechanics energy