Tuesday, June 15, 2021

[idcbhflx] relativistic velocity (squared) as a function of energy

special relativity:

v^2 = (c^2*E^2 + 2*c^4*E*m)/(E + c^2*m)^2

take square root to get velocity.

Taylor series in powers of E:

v^2 = 2*E/m - 3*E^2/(c^2*m^2) + 4*E^3/(c^4*m^3) - 5*E^4/(c^6*m^4) + 6*E^5/(c^8*m^5) ...

the first term matches Newtonian KE = (1/2)*m*v^2 .

expanding around Infinity (negative powers of energy):

v^2 = c^2 + 0 - c^6*m^2/E^2 + 2*c^8*m^3/E^3 - 3*c^10*m^4/E^4 + 4*c^12*m^5/E^5 ...

v = c when infinite energy.

both series appear to have simple form.



(double square brackets are syntactic sugar for the Part function.)

inspired by various xkcd what if? posts on the theme of, what if we added more energy?

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