Like Ork says. There is an aeromodelling discipline called F1D, and they build planes that fly, well for a long time. The world record is OVER AN HOUR on a rubber band weighing 0.6 grammes !!!!! From www.fai.org/aeromodelling/F1+synopsis+indoor The World Championships class F1D permits rubber motors of up to 0.6 gram in an aircraft with a minimum weight of 1.2g the without rubber motor. Dimensional limits are maximum wing span of 550mm, maximum chord of lifting surfaces 200mm, and maximum tail span 450mm. Models are usually made of a balsa frame covered with a lightweight transparent covering, either microfilm or plastic. Flights of up to 40 minutes are possible with F1D models. There are two other rubber powered classes for smaller models: The record is
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Start by building a glider that can stay in the air for a long time. Then add rubberband and propeller to lengthen that time. Note, however, that when the power runs out the propeller will add drag, so there's going to be some trade-off in terms of propeller (and rubber band) size and weight and shape, and a mechanism which will let the propeller "freewheel" when not powered will probably be helpful Websearch will probably find some good designs. The fanciest I've seen described involved ultra-light frameworks with wing and propeller surfaces of VERY thin plastic, stretched to make it rigid enough not to flutter -- I think some of them made the prop and wings out of wire dipped in a liquid plastic that formed a film almost as thin as a soap bubble, extremely light and just barely strong enough to hold up to flight.