A spaceship at rest in a certain reference frame S is given a speed increment of 0.50c. Relative to its new rest frame, it is then given a further 0.50c increment. This process is continued until its speed with respect to its original frame S exceeds 0.999c. How many increments does this process require? (Answer: 7)

Answer 1

v(n+1) = {v´+v(n)}/{1+(v´•v(n)/c²}
v2= {v´+v1}/{1+(v´•v1/c²}=
={0.5c+0.5c}/{1+(0.5c•0.5c/c²} =0.8c.
v3={0.8c+0.8c}/{1+(0.8c•0.8c/c²}...0.929 c
v4=...=0.97561c,
v5=...= 0.9918c
v6=...=0.99726c,
v7=...=0.99909c > 0.999c