Answer
The speed of the rocket relative to the earth is $~~0.345~c~~$ opposite to the direction of the projectile.
Work Step by Step
Let $u'$ be projectile's velocity relative to the earth reference frame.
Let $v$ be the rocket's velocity relative to the projectile.
Let $u$ be rocket's velocity relative to the earth.
We can find $u$:
$u = \frac{u'+v}{1+\frac{u'v}{c^2}}$
$u = \frac{0.90c-0.95c}{1+\frac{(0.90c)(-0.95c)}{c^2}}$
$u = \frac{-0.05c}{1-0.855}$
$u = -0.345~c$
The speed of the rocket relative to the earth is $~~0.345~c~~$ opposite to the direction of the projectile.
