Answer
The shuttle should be moving at a speed of 7460 m/s when it releases the satellite.
Work Step by Step
Let's use 6380 km as the Earth's radius. Then the orbital radius $r$ is 6380 km + 780 km which is 7160 km. The force of gravity provides the centripetal force for the satellite to move in a circle around the Earth.
$\frac{mv^2}{r} = \frac{GMm}{r^2}$
$v = \sqrt{\frac{GM}{r}}$
$v = \sqrt{\frac{(6.67 \times 10^{-11} ~N\cdot m^2/kg^2)(5.98 \times 10^{24} ~kg)}{7.16\times 10^6 ~m}}$
$v = 7460 ~m/s$
The shuttle should be moving at a speed of 7460 m/s when it releases the satellite.