Answer
The speed of the rocket piece relative to the satellite was $~~38,240~km/h$
Work Step by Step
We can find the speed of each object:
$\frac{mv^2}{r} = \frac{GMm}{r^2}$
$v^2 = \frac{GM}{r}$
$v = \sqrt{\frac{GM}{r}}$
$v = \sqrt{\frac{(6.67\times 10^{-11}~N~m^2/kg^2)(5.98\times 10^{24}~kg)}{6.37\times 10^6~m+7.0\times 10^5~m}}$
$v = 7.511~km/s$
We can find the relative speed if the collision was perpendicular:
$v = \sqrt{(7.511~km/s)^2+(7.511~km/s)^2} = 10.622~km/s$
We can express this speed in units of km/h:
$v = (10.622~km/s)\times \frac{3600~s}{1~h} = 38,240~km/h$
The speed of the rocket piece relative to the satellite was $~~38,240~km/h$
