Assuming no air resistance, why does a projectile launched horizontally at 8 km/s not strike earth's surface?

orbital velocity of an object around earth is given by

here we can assume that object is moving at distance "r" from the center of earth

and net gravitational force on the object will provide it centripetal force to revolve in the circle

so here we have

now if the object is projected from surface of earth

then we have

now if we plug all values in above equation

by solving above equation we have

so when object is project horizontally with speed 8 km /s then it will not hit the surface but it will circulate around the earth at constant speed

the independent variable, or the one that is intentionally manipulated is the speed.