CM_trains.pdf

1)
A' goes 3->5 in 0(A) to 4(C); so 2 gamma = 4, gamma=2
E goes 6->8 in 3(C') to 7(A'); so 2 gamma = 4, gamma=2
all four units of AE fit between A'(3) and C'(3), so 4 shrinks to 2 so gamma=2
2^2=1/(1-beta^2); 4-4 beta^2=1, 3=4 beta^2, beta=sqrt(3)/2

2)
beta=dx''/(2*gamma*c)
1=dx''*beta*gamma/c; dx''=c/(beta*gamma)

beta=(c/(beta*gamma))/(2*gamma*c)
beta^2 gamma^2=1/2
beta^2=(1/2) (1-beta^2)
(3/2)beta^2=(1/2)
beta=1/sqrt(3)

gamma=1/sqrt(1-beta^2)=sqrt(3/2)
so dt''=2 sqrt(3/2) (time dilation)

3)
(1/sqrt(3)+1/sqrt(3))/(1+1/sqrt(3)^2)=2/sqrt(3)/(4/3)=sqrt(3)/2