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Combustion engines are tried and true, and however angry they might look and sound in a top-fuel dragster or space rocket booster, the combustion process of oxidizing fuel in air is relatively slow and predictable. Detonation, on the other hand, is as chaotic and destructive as it sounds. It's how most bombs work; you take an explosive fuel and hit it with a jolt of energy, and the chemical bonds holding each molecule together break apart, releasing wild amounts of energy in a shockwave that expands at supersonic speed.
NASA, along with many other groups, wants to harness these explosions for a couple of key reasons. Firstly, detonation engines have a considerably higher theoretical level of efficiency than combustion engines, perhaps as much as 25%; they should be able to produce more thrust using less fuel and a smaller rocket. In the engineering and economics of space flight, that means cheaper launches, more billable payload, and greater distances.