Answer
The speed of the rocket piece relative to the satellite was $~~54,080~km/h$
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 head on:
$v = (2)(7.511~km/s) = 15.022~km/s$
We can express this speed in units of km/h:
$v = (15.022~km/s)\times \frac{3600~s}{1~h} = 54,080~km/h$
The speed of the rocket piece relative to the satellite was $~~54,080~km/h$
