Customized Digital Clock(two power sources)
Maybe you could make your idea with the AND gate work,but it's not necessary.
There is a simpler way than that:both transistors and relays can switch on a higher load current with a smaller current.Here is an example:  







simulate this circuit &ndash; Schematic created using CircuitLab
The transistor will be switched on only when there will be current applied to the base(It comes from the blue and black wires.Note that the positive one of those wires will be connected to the base and the other to the emitter),switching the load(motor) with the help of the external 3V battery.You will have to know the current needed for the load and then calculate the base current accordingly using the NPN's hFE to turn it fully on(or how much you need) in order to choose the right resistor values.Let me know if you need more information.