Orbital velocity is about Mach 25. You would need to be going much faster than that, if your only power is applied in the atmosphere (you'd need enough additional speed to get above the atmosphere, and enough to overcome air resistance. And after the cable lets go, the the orbit will be elliptical, which means that at the end of one orbit you come right back to the launch site. So you need to bring enough fuel with you to flatten out the orbit once you are at the highest altitude. Oh, and then you have the weight and tensile strength of the cable, so similar issues to that faced by building a space elevator.

A better idea is the launch loop, but again you have to deal with cable weight and tensile strength.