I had a car do exactly that
Except I had a 45 minute drive to work so it was a non problem. Whats happening is your alternator isn't "starting." Alternators are esentially self starting, and yours isn't. Part of that effect is the dash "alternator" light, check that, it should come on with the ignition on and the engine off.
A trick I did was after everything warmed up a little reving the engine up. That helped it "start" sooner.