Question

At what values of the ratio of the kinetic energy to rest energy can the velocity of a particle be calculated from the classical formula with the relative error less than `epsilon=0.010`?

Answer 1

Relativistically
`(T)/(m_0c^2)=((1)/(sqrt(1-beta^2))-1)=1/2beta^2+3/8beta^4`
So `beta_(rel)^2~~(2T)/(m_0c^2)-3/4(beta_(rel)^2)~~(2T)/(m_0c^2)-3/4((2T)/(m_0c^2))^2`
Thus `-beta_(rel)=[(2T)/(m_0c^2)-3(T^2)/(m_0^2c^4)]^(1//2)=sqrt((2T)/(m_0c^2))(1-3/4(T)/(m_0c^2))`
But Classically, `beta_(cl)=sqrt((2T)/(m_0c^2))` so `(beta_(rel)-beta_(cl))/(beta_(cl))=3/4(T)/(m_0c^2)=epsilon`
Hence if `(T)/(m_0c^2)lt4/3epsilon`
the velocity `beta` is given by the classical formula with an error less than `epsilon`.