Ask A Scientist Weather Archive

Judge

I needed to find the terminal velocity of a hail stone and the impact 
force. And was wondering if there was any way you could show me step by step how to solve
this.
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Judge,

The equation for terminal velocity applies for a perfect
sphere, which a hailstone isn't necessarily, but it is the
best that we can do.

Vt=(0.2222*g*(dh-da)*(r**2)/n

where * stands for multiplication, ** stands for squared,
/ stands for division, g is the acceleration of gravity (980 cm/s**2),
dh is the density of the hail (0.917 g/cm**3 for ice), da is the
density of air (approximately 0.0012923 g/cm**3),
r is the radius (half the diameter) of the hailstone in cm, and n is
the dynamic viscosity of air near the Earth's surface (approximately
0.00018 g/cm/s).

For a hailstone 1 inch in diameter (1 inch multiplied times 2.54
cm per inch gives a diameter of 2.54 cm), it has a radius r of 1.27 cm.
Squaring this gives 1.613 cm**2.

Now we can work the equation:

0.222 * 980 * 0.9157 * 1.613 / 0.00018 = 1,814,721.9 cm/s

or translating to some other units, (using 1 cm/s = 0.03281 ft/s)
gives a terminal velocity of 59,541.03 ft/s
or (using 1 cm/s = 0.02237 mph)
gives a terminal velocity of 40,595.3 mph.

Obviously, the hailstone will not reach this terminal velocity
(it would melt from friction at this speed).  Our calculation of
terminal velocity does not take into consideration that
the hailstone usually is hindered from falling by being in a thunderstorm
updraft during most of it's formation and fall to the ground, and by
collisions with other hailstones or water droplets as well, slowing
it's fall.

But, let's look at the terminal velocity of pea size hail, say 0.3 cm
in diameter, giving a radius of 0.15 cm. Squaring this gives
0.0225 cm**2.  Using this in the equation gives 25,313.9 cm/s, or
830.5 ft/s, or 566.3 mph.  This is also quite unlikely.  At this speed
the hailstone should make a nice hole in the ground when it hits.  I have
never seen a small hailstone (or a large one for that matter) make a
hole in the ground, so I can conclude that a hailstone never has a chance
to come anywhere close to reaching it's terminal velocity.  Still, it's
an interesting calculation.  Not knowing the fall velocity, we can't
calculate the impact force.

Hailstone speed (combined with large size) and impact force can be
significant however, as I observed sizable dents in my car and in the
aluminum siding on my house after a hailstorm of 1.25 inch diameter
hail several years ago.


David R. Cook
Atmospheric Research Section
Environmental Research Division
Argonne National Laboratory
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