Objects Falling Through Air
Question:
An arguement I had last night (fuelled by liquor of course), lead me to your website in search of answers. The question possed was "if I drop and cow and a concrete block out of a 20 storey building, which hits the ground first?". Having studied physics and mechanics at University, my response was that they both land together. The explanation given is that, assuming air resistance is similar for both objects, mass of the object has no influence over it's acceleration (and therefore the time it takes to hit the ground). My friends, however, failed to believe this explanation and possed the further question: "what if you have a balloon filled with water and one filled with air?". In this instance, it's hard to image that a balloon filled with air will fall as fast as a balloon filled with water. Can you explain why this might be so?
Answer:
The key is in the nature of air resistance. Air resistance is not a constant force but dependent on velocity. In fact a decent approximation of the force of air resistance, Fa, is that Fa=s*A*d*v^2 where s is an experimentally determined constant, A is the cross-section area of the falling object, d is the density of the fluid through which it is falling, air in our example, and v is the velocity. A falling object then will accelerate downward, increasing its velocity until Fa=mg, the mass times the acceleration of gravity. At this velocity the net force on the object is zero so velocity stops increasing. This is the terminal velocity of which you may have heard.
Objects with cross-section large in proportion to their mass reach their smaller terminal velocity sooner than objects whose cross-section is small in proportion to their mass. Therefore with objects with similar cross-section the heavier one reaches the ground first. The extreme example of this is the water droplets that make up a cloud. Although the cloud may have a combined mass of several thousand tons, the aggregate cross-section of the droplets is enormous, resulting in a terminal velocity so small that it is less than the velocity of thermal convection currents in the air and the cloud actually grows upward in altitude.
If the terminal velocity for both objects is substantially above that velocity achieved over the falling distance then objects of quite different mass will hit the ground very nearly the same time since mg remains the dominant force governing their motion.