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The speed of one rocket relative to the other is 8.5 cc.
Let s be the frame at rest relative to the earth. Let the spaceship moving to the left with respect to the earth at speed [tex]v[/tex] be the particle in the terminology. Then the velocity of the spaceship relative to each other is simply the velocity of the second spaceship.
That for speeds small compared with the speed of light the relativistic velocity addition law gives the usual Newtonian result, whereas for speed near c the result is not what has been expected classically.
[tex]v' = 0.96cc\\\\v= 0.84cc\\\\\\v' = \frac{v+V}{1+ \frac{vV}{c^2} } \\\\v' = \frac{v+V}{ \frac{c^2+ vV}{c^2} }\\\\v' = \frac{(v+V) c^2} {c^2+ vV}\\\\\\v'c^2 + c^2v = V (c^2 - v'v)\\\\V = \frac{v'c^2+c^2v}{c^2 - v'v} \\ \\ = \frac{0.96 * 9* 10 ^1^6 + 0.84*9*10^1^6}{9*10^1^6 - 0.96 * 0.84} \\\\ =\frac{16.2}{1.9}\\ \\= 8.5 cc[/tex]
Therefore, the speed of one rocket relative to the other is 8.5 cc.
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