Air Speed is the relative speed between the air surrounding the aircraft and the aircraft itself.
Ground Speed is the relative speed between the aircraft and the ground.
For example, consider an aircraft flying at a speed of 100 KTS and having a nose-wind of 30 KTS. The speed on the aircraft's gauges is showing 100 KTS as the aircraft is really flying at that speed within the block of air. But, the entire block of air is moving backwards at 30 KTS, hence the aircraft's ground speed is in fact only 70 KTS.
Airspeed is used in any reference to the aircraft's aerodynamic behavior. For example, an aircraft may stall if its airspeed decreases bellow the minimal airspeed regardless of its ground speed.
Ground Speed is used mainly for time calculations (how long would it take to get from one place to another).
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