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Why is, for example, kinetic energy proportional to velocity squared? One might have grown accustomed to this over the years, but are there any simple explanations of this fact to a beginning student of physics?

I have thought about work (W=Fd) and the fact that it is not invariant under Galilei Transformation (d is not invariant). This is contra intuitive for the reason that we would believe that the energy consumed by an accelerating spacecraft would be independent of the observer. However, if you look at the kinetic energy of the rocket fuel (perhaps dropping in speed when leaving the rocket engine) there is no paradox. In fact one can show (I have shown it in a simple case below) that in a system of particles, under the valid assumption that momentum is conserved, the energy difference from one moment to the next will be independent of the observer. (This would also apply to the spacecraft of course.) The conclusion is that the energy change (to or from kinetic energy) is independent of the observer.

*If energy change is Galilei invariant this would strongly suggest that the idea of velocity squared is correct*.

However, I don’t know if it is possible to explain this to a school student. And it gets worse if you look at a car accelerating. Here the energy is transformed into rotational energy in the wheels, which in its turn is transformed into kinetic energy of the car and rotational energy of the planet. But I guess it is possible (doing the calculations was too much for me) to reduce this to the particle interacting case, the earth being a big particle and the car being a small. I wont evaluate my thoughts about this right here, but can say that crucial to analyzing this situation is that you can never fix the inertial system to the ground, since the whole planet will be affected. In school books they never discuss this fact, of course.

And yet, although I know that the acceleration of a car does not violate Galilei invariance, and although I would agree that the high speed of a car is “energy in itself” and this is why energy is proportional to velocity squared, it is still not obvious to me why it takes more fuel to accelerate from 50 km/h 100 km/h than from 0 km/h to 50 km/h. I think my problem is that in physics you always lock everything into an inertial frame, but in reality there are no obvious inertial frames. And this is why I would falter if I where to explain why this is so to someone else. Are there any good physicists out there who can explain why I never seem to reach a full understanding of this energy formula?

/Order

PS. I know the question of energy has been in focus before at this forum. If anyone has a really good link where all my questions have an answer, it is ok to report it here. DS.

PPS. I wont give a full mathematical explanation of why energy change is galilei invariant, but one can look at a simple example: suppose you have a ball of mass 2m moving with a velocity v (this fixes the inertial frame in one dimension). Then suddenly this ball is split into two equally massive parts. So [tex]m_1 = m[/tex] and [tex]m_2 = m[/tex]. Particle 1 is boosted with a velocity h. Therefore, in order to concerve momentum, particle 2 is boosted with a velocity -h. The splitting energy then is

[tex]\Delta E = \frac{m_1 (v + h)^2}{2} + \frac{m_2 (v - h)^2}{2} - \frac{m_1 v^2}{2} - \frac{m_2 v^2}{2}= \frac{m}{2} (-2 v^2 + 2 v^2 + 2 h^2) = m h^2[/tex]

Since the splitting energy is independent of the initial velocity v it is Galilei invariant. Therefore logic is not violated.

One can extend this calculation to involve different masses and different initial velocities of the two particles. But the interaction energy will still be Galilei invariant. DDS.