KE=1/2mv^2 revisted!thank all of you guys for answering my prior question on it! I hate to beat a dead horse..

Question

I emailed the author of lightandmatter.com regarding his textbook Simple Nature. This is an excerpt of my email

""I have an objection to your statement on page 38 bottom paragraph. In reference to the kinetic energy theorom, KE=1/2mv^2 , you state that "what about the factor of 1/2 in front? It comes out to exactly 1/2 by the design of the metric system."

That statement is not accurate since the Kinetic energy equation is fundamentally idependent of the unit system used. Whether we were to use the old english system, the metric system, or any other system the 1/2 would still be required for the equation to hold. I believe you meant that if you were to covert the old English sysetm to the metric system the factor that you stated is required.""

He replied by stating that 1/2 was indeed arbitrary.

Now I know He is 100X smarter than me so if any of you hit his page and read his section of the book tell me What am I getting wrong. Difference of semanantics?What am I missing?

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Answer 1

Here's the link to the HTML version of Benjamin Crowell's textbook on physics, the "Conservation of Energy" chapter. The excerpt regarding kinetic energy is as follows:

"The proportionality factor equals 0.5 because of the design of the metric system, so the kinetic energy of a moving object is given by KE = 1/2 m v^2"

All right, the short and simple reply to this is that Crowell is 100% wrong. There is no further debate or mystery about this. All of the others here and previous times this question has been posted have given good answers. The coefficient of 1/2 has nothing to do with any system of measurement, it's an inherent mathematical artifact contained in the definition of kinetic energy. As I have answered before, in the approximation (principle of correlation), special relativity also gives this result, without resorting to Newton's methods of calculus in deriving the expression for kinetic energy. In relativity work, geometrized unit system is frequently used to simplify things, making it independent from any arbitrary "metric" system (as Crowell calls it), and STILL, in the approximating case, the relativistic expression for kinetic energy reduces to 1/2mv^2 where v is small. If this 1/2 factor was not there, relativity physicists would be seriously scratching their heads as badly as they did after discovering only about half of the expected amount of solar neutrinos were arriving at earth! (That problem was solved with the discovery of oscillating neutrino mass).

Answer 2

Everyone is right here. Of course the 1/2 is there because of the way energy is defined, but taking the energy from a different metric or non-metric system than m and v, you'll end up with a totally different constant:
Take for instance E in eV (electronvolt), m in g and v in m/s you'll end up with a completely different factor than 1/2. In this respect the "author" can be said to be right.

The unit of energy is normally always chosen to match the rest of the units in a logical way. The 1/2 is a logical factor here and should be there for physics to be a logical theory. The unit of energy is therefore always chosen to reflect this 1/2, which is arbitrary or not, depending on the way you look at it.

For physics to be a logical theory, the 1/2 is NOT an arbitrary constant. But c=299,792,458 metres per second is an arbitrary constant, totally depending on the choice of units for length and time. Why not make c=1 lengthunit/timeunit? This would also be arbitrary, depending on how you look at it, but it makes some calculations easier. E=m*c^2 becomes E=m etc.

Answer 3

The 1/2 is necessary because of the way we define kinetic energy with respect to velocity and momentum. Momentum is essentially the velocity derivative of kinetic energy, the derivative of kinetic energy with respect to velocity. In order for this idea to be true, the equation for kinetic energy must have a 1/2 in it.

Think of this:

p = mv
dk/dv = p
dk = p dv
dk = mv dv
k = 1/2 m v^2

Answer 4

The kinetic energy theorem is derived from Newtons laws, and the laws of calculus. The one half comes from an integration in the derivation. It has absolutely nothing to do with the metric system.

Answer 5

The equation KE = 1/2mv^2 is got by integration. The 1/2 comes from the "rules" of calculus.

Mention it to your Maths teacher, who will know the most appropriatre way to explain to you. (I don't know how much Maths. you've done.)

Answer 6

Me again.

The 1/2 you speak of is arbitary in that it makes the rest of the equation more true than any other number. As I tried to say in my last answer, given what we have and where our knowledge lies, 1/2 is the best guess we have and it makes the equation as true as we can reasonably make it.

In the future as our knowledge expands, we may assign something else or even decide that our current equation is not good enough. After all, look at what the Huble telescope is teaching us about our current physics.

Answer 7

When he says it is arbitrary, maybe he means that you are right, and he means to say that the 1/2 is correct for an arbitrary system of units.