High-altitude free fall

Abstract

The problem of an object falling from high altitudes where the variation of atmospheric pressure cannot be neglected is investigated. The equation of motion for the variation of the velocity of the object as a function of altitude is solved exactly. The results show that, unlike an object falling in a uniform atmosphere whose speed monotonically increases and approaches the terminal speed, the speed of a high-altitude falling object first increases, goes through a maximum, and then decreases and approaches the terminal speed from above. The results also show that if the initial altitude of the object is greater than a critical value, the object always strikes the ground with a speed that is higher than its terminal speed by a finite value, in contrast to the case of a freely falling object in a uniform atmosphere.