Some insects can live for substantial periods without a head, although they will usually die of dehydration or starvation within a day or two (cockroaches may live even longer). This is because their breathing and other autonomic functions do not require direct control by the brain, and unless they lose substantial blood and fluids, many of their organs will continue to function.
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A grasshopper can live several minutes without air. Grasshoppers will eventually die if they do not receive a fresh air supply.
Grasshoppers and Locusts are the same insect they are called locusts when they swarm. Swarms can fly up to 300 Km in a day. Thus they can fly for up to 12 hours.