A space vehicle is orbiting Earth in a circular orbit with a radius of 10,300,00

We have: The force of gravity is:
F=G*M*m/d2   (M is earth mass, m is the vehicle mass and d is the distance between earth and vehicle)

Since we don't know the m, we will find the gravity acceleration instead:
g=F/m = G*M/d2 =  6.67*10-11* 5.97*1024/(10.3*106)2=39.82*1013/(10.32*1012) =   398.2/106.09=3.7534m/s2

The centripetal force is equal to the gravity force. The centripetal force is: Fc=m*v2/r
And the centripetal acceleration is equal to the gravity acceleration and: ac= Fc/m = v2/r = v2/10.3*106

So, to solve the exercise you just need to do this:
g=ac <=> ... <=> 3.7534 = v2/(10.3*106) <=> v2= 3.7534*10.3*106 = v <=> v2 = 38.66*106 <=> v = 6.2177*103m/s

This is the exact velocity required so that the vehicle will do a circular orbit with the given radius around earth. A minimum increase to this velocity will make the vehicle escape earth's gravitational field. So, the answer should be this: v = 6.2177*103m/s

Please double check the calculations.