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.

Mathematica:

v2=Solve[e==m*c^2*(1/Sqrt[1-(v2/c^2)]-1),v2][[1]][[1]][[2]]

Series[v2,{e,0,5}]

Series[v2,{e,Infinity,5}]

(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?

## No comments :

Post a Comment