Larry Weinstein, Column Editor
Old Dominion University, Norfolk, VA 23529;
[email protected].
Fermi Questions
Solutions for Fermi Questions, January 2016
w Question 1: Rain
time is
How much water will hit us if we walk a kilometer (or
mile) in a heavy rainstorm? Assume there is no wind.
Answer: The amount of rain hitting us will depend on
our walking speed, our surface area, and the rate of rainfall. If we walk infinitely quickly, then we will intercept all
the rain in the volume swept out by our frontal area. If,
like most people, we walk at finite speed, then we will still
intercept all the rain in a volume swept out by our frontal
area (except that instead of the volume being parallel to
the ground, it will be at an angle determined by our walking speed and the speed of the raindrops) and we will also
intercept the rain falling on our heads and shoulders. (See
http://www.bbc.com/news/science-environment-18901072
for more details.)
We can walk slowly (1 m/s or about 2 mph) or quickly
(2 m/s or about 4 mph). Our frontal surface area is about
1 m2 (2 m by 0.5 m) and the surface area of our head and
shoulders is about Atop = (0.5 m)(0.25 m) < 0.1 m2.
A heavy rain falls at a rate of about 2 inches (5 cm) per
hour. That is a rate of
We can determine the speed of falling rain in one of
several ways. In a 30 mph wind, rain falls at about 45o,
indicating that its terminal velocity is about 40 mph.
Alternatively, the rear window of a sedan remains dry at
speeds over about 40 mph. If the rear window is at 45o,
then the terminal velocity of the rain is also 40 mph, or
about 20 m/s. This implies that the fraction of the air occupied by falling rain is
Even in a very heavy rainstorm, only 0.5 parts per million
of the air is occupied by water.
The volume swept out as we walk to our destination is
V = Ad = (1 m2)(103 m) = 103 m3
and the water contained in that volume is
Vw = fV = (5310–7)(103 m3)
= 5310–4 m3 = 0.5 L.
The rain falling on our head and shoulders during that
The Physics Teacher ◆ Vol. 53, 2015
Even at a fast walk, just as much rain will hit our head
and shoulders as hits our front. Thus, we will be hit by a
total of 1 L of water.
Yes, we will be very very wet. But you knew that before
we did this calculation.
Copyright 2016, Lawrence Weinstein.
w Question 2: Drying
How long will take to dry all of our clothes (assuming
we walked a mile in the rain without umbrella or rain
jacket) with a hair dryer? (Thanks to Alex Godunov of
Old Dominion University for suggesting the question.)
Answer: After walking home in the rain, our clothes
are soaked. The time to dry them with a hair dryer will
depend on the heat output of the hair dryer, the water
remaining in the clothes, the heat capacity and latent
heat of water, and the efficiency of the hair dryer in
transferring heat to the clothes.
A hair dryer will typically have the same maximum heat
output as other appliances that plug into normal wall
outlets, such as toasters, microwaves, and space heaters.
If you don’t remember the output of one of these, you
might remember that this output is limited by the typical
110 V, 20 A circuit breaker. In either case, we’ll use P =
103 W.
Although 1 L of rain has hit us during our long walk
home, not all of it remains in our clothing. The amount
of water remaining will be more than 1% and less than
100%, so we will estimate 10%. This will obviously
depend on the clothes we wear. A heavy cotton shirt
and blue jeans will retain much more water than a thin
synthetic shirt and pants. We can adjust our estimate according to the clothing we wear. For now we will use m =
(103 g)(0.1) = 102 g.
While we can easily look up the heat capacity and latent
heat of water, it is useful to be able to estimate it. A calo-
rie is defined as the energy needed to raise the temperature of 1 g of water by 1 oC. (A BTU is similarly defined
as the energy needed to raise the temperature of 1 lb of
water by 1 oF, but that is less useful here.) This means
that it takes 100 c or 400 J to raise the temperature of
1 g of water from freezing to boiling. Since it takes several times longer to boil off all the water in a tea kettle
than to bring it to boiling, the latent heat of evaporation
is several times larger, or about 103 J. In actuality it is L
< 23103 J/g, so our crude estimate is not too bad.
The efficiency of the hair dryer in transferring heat to
the clothes will be more than 1% and less than 100%
so we will estimate e = 10% = 0.1. This includes both
the heat transferred to the ambient air rather than the
clothes and our unwillingness to risk melting synthetic
clothing.
Now we can put it all together. The time needed will be
or about 30 minutes. Heavy cotton clothing will take
more time, and light synthetic clothing will take less.
Since it takes about a minute to dry our hands using
rest-room hand dryers, this estimate is not crazy.
Washing machines have a spin cycle to remove water
from clothing to reduce the time and energy needed
for drying. Mountain huts for hikers often have drying
rooms because they don’t usually provide hair dryers.
Copyright 2016, Lawrence Weinstein.