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If x=0 at t=0 and x'=0 at t'=0 and they are initially set to have t=t'=0, this doesn't mean that t'=t for all t or t' (which ever you are taking as the independent variable). For an instant their clocks will agree and then they will not. The gamma factor tells you that immediately. It's like saying :
"Let T = 2(t-t0). To set origins equal we put t0=0, so initially, at t=0 T=0. Therefore t=T for all t."
Obviously false but it's what you're doing.
You do realise these are exercises given to school children? I help teach 'Relativity and Motion' at my university and time dilation is included. Of course they don't do it with scalar equations, they aren't up to speed on using tensor calculus yet, they are only 1st years.
"Let T = 2(t-t0). To set origins equal we put t0=0, so initially, at t=0 T=0. Therefore t=T for all t."
Obviously false but it's what you're doing.
You do realise these are exercises given to school children? I help teach 'Relativity and Motion' at my university and time dilation is included. Of course they don't do it with scalar equations, they aren't up to speed on using tensor calculus yet, they are only 1st years.
However, it does not help to explain away the fact the one particle in one state in k, described by F' = ma' is represented by two different states in K, namely, F = ma and F = beta^3ma, if we believe both the “Principle of Relativity” and the Lorentz transformations are physically correct. A single particle in a given state in a given system cannot be represented by a particle having two different states in another system, don’t you agree?