Answer to the Question 12/01
BAD WEATHERThe question was:
One winter day a weatherman in New York said that the snow was melted by the recent rainfall and that's why there were floods. There were about 30 inches of snow and about 1 inch of rain. Is the weatherman correct in his assertion?
(03/02) The problem has been answered correctly by Will Brunner (6/12/2001) from the University of Colorado at Boulder (US) (e-mail brunner@terra.colorado.edu).
Answer: The weatherman was wrong.
The solution:
If you ever looked at a snowflake, or kept snow in your hand - you surely know how tenuous it is. The density of snow is approximately 0.1 g/cm3, although it depends on temperature and other weather conditions and maybe as large as 0.2 and as small as 0.02. Consequently, most of the snow is simply "air", and 30" of snow corresponds to about 3 inches of compressed material (ice). Consequently, even if 1" of rain could melt all the snow we would be faced with 4" of water. However, we need to examine whether the rain could actually melt all the snow. Let us assume that it was really warm rain (in the middle of the winter) of say 20 degrees C. Each cm3 of rain the would cool to 0 degrees and release 20 calories of heat. The latent heat of ice is 80 cal/g, and consequently 1" or rain could melt 1/4" of ice, leading to amount of water equivalent to 1.25". Certainly not enough for a flood.
Comment: Bill Bruml (13/1/2002)) (e-mail bruml@alum.mit.edu) pointed out two important points that may modify the above solution or even make it invalid:
1. Snowflakes have a very large surface (relative to their bulk). This may effectively decrease the latent heat.
2. If the snow was really not snow, but "slush", i.e. matrix of frozen material holding water - then the heat will only be required to melt the matrix, and thus much more water can be generated.
Comment: (26/12/01) Luca Visinelli - an Italian student (e-mail luca.visinelli@libero.it) pointed to an additional potential source of energy: the kinetic energy of falling drops is converted to heat. However, for rain drops falling at say 10 m/s velocity, the kinetic energy of 1 gm of rain is 0.05J=0.01cal. Completely negligible amount! (The velocity of the drop is a result of compensation between its weight and air friction and, consequently, depends on its radius, but for any reasonable value of drop size, the kinetic energy is still too small.)
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