What are the extra terms in the k.e expression

[Ian]

Can someone please tell me what the first six terms are in the classical expression for kinetic energy.
Thanks,
Ian.

 

[suyver]

1) The kinetic energy of what?
2) Why did you post this in the general math forum?

 

[Njorl]

Start with the relativistic total energy (rest and kinetic):

E²=(mc²)²+p²c²

p=mv

substitute and simplify to E=mc²(1+v²/c²)^1/2

then use the binomial theorem. The first term of the binomial expansion will be the rest energy, the next term will be the commonly used (1/2)mv². The second through seventh terms are the ones you want.

Njorl

 

[suyver]

Maybe it is a matter of semantics, but I wouldn't call that the "classical expression for kinetic energy", rather the "relativistic expression for kinetic energy". I guess that it's just a matter of taste...

 

[Njorl]

Originally posted by suyver
Maybe it is a matter of semantics, but I wouldn't call that the "classical expression for kinetic energy", rather the "relativistic expression for kinetic energy". I guess that it's just a matter of taste...

I agree. I don't think there is a standard terminology for the next few terms of the expansion. I suppose they could be called the first through fifth order corrections to the classical formula for kinetic energy.

Njorl

 

[ahrkron]

Originally posted by Ian
Can someone please tell me what the first six terms are in the classical expression for kinetic energy.
Thanks,
Ian.

They do not correspond to any named quantity in classical physics.

This needs not be surprising. You can write any function as a linear combination of other functions in many ways (much in the way you can use different coordinate systems for describing vectors).

It turns out that, when you expand the K.E. in powers of v/c, you gain some understanding of why things look like classical mechanics when v/c<<1, but the (infinitely many) higher-order terms do not have a wide enough use to get them a name of their own; actually, had relativity not prompted this peculiar expansion of KE in terms of powers of (v/c), those terms would probably never had shown up elsewhere.

 

[Integral]

I guess for the most part I would simply call them, ignored!