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How can the kinetic energy equation be intuitively understood?

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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.

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In short, momentum is vector and kinetic energy is scalar.

$$\vec p = m \vec v \tag{1}$$ $$T = \frac{1}{2}m \vec v^2=\frac{\vec p }{2m} \tag{2}$$

Two momenta in opposite direction is $0$. Total energy and momentum both are conserved, let total energy =kinetic energy + potential energy. Here kinetic energy changes its form (it mostly converts to other energy since energy can never be destroyed) while momentum is conserved body to body. In equation (2) I had shown that kinetic energy is mostly relevant to momentum (but they aren't same), the core difference here is one is scalar another is vector. Let's look at their units (other prefers dimension but I like units), units of momentum is $kgm$ while units of kinetic energy is $kgm^2$. Kinetic energy is movement energy of an object. Think that, in a system some kind energy is converted to kinetic energy, so the energy will make particles move in that system.

If two objects move at the same speed then the more massive one has more quantity of motion and if two objects have same mass then the faster one has more quantity of motion.

Kinetic energy is energy of moving object.

An integral form might be more helpful to understand the difference,

$$\vec p = \int \vec F \ \mathrm {dt}$$ $$W=T=\int \vec F \cdot \mathrm {d\vec s}$$

Further reading :

  1. http://www.differencebetween.info/difference-between-kinetic-energy-and-momentum
  2. https://physics.stackexchange.com/q/16160/315444
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2 comment threads

Thanks, and I'm sure that makes it intuitive to a physicist, but it's not quite what I was hoping for... (1 comment)
This answer doesn't address the question. (1 comment)

Comments on How can the kinetic energy equation be intuitively understood?

Thanks, and I'm sure that makes it intuitive to a physicist, but it's not quite what I was hoping for...
gmcgath‭ wrote 10 months ago:

Thanks, and I'm sure that makes it intuitive to a physicist, but it's not quite what I was hoping for. Something like the discussion of the effects of a collision vs. the velocity going into it might provide more of a feeling for why kinetic energy is proportional to the square of the velocity.

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