Hail! Hail! To Michigan!

[Title from the University of Michigan fight song, which Steve couldn't stop singing when I was accepted to the graduate program.]

A couple of weeks ago, we had a sudden hailstorm while I was cooking dinner, and the kitchen skylight made some incredible (and alarming) sounds:

I started to wonder whether I was in danger of being showered with glass, and while the skylight (and I) survived the storm, I thought I'd take a look at what the chances were for future shattering.

Hailstones can form in tall thunderclouds with lots of air movement, where water drops can rise into cooler regions and freeze, then fall partway to collect more water. This repeats several times before the hail falls. If it falls far enough, it will reach terminal velocity, the speed at which the force from gravity pulling down balances the air resistance slowing it down:

where m is the mass, g the gravitational acceleration, ρ is the density of air, A is the cross-sectional area of the hailstone, and Cd the drag coefficient, which depends on the shape of the object. Since I'm a physicist, I'll assume the stones are spheres with uniform density. Then we can express the mass and area in terms of the radius of the hailstone, and use the tabulated value for the drag coefficient:

Plugging these into the velocity equation above, we find

I wanted to check whether this was reasonable, so I found a paper from the University of Wyoming with this plot:

Figure 1

If we plot our function over the same range of diameters, the velocities match incredibly well, given the simplifying assumptions we made:

Now that the velocity is settled, we can look at how much energy the hailstones carry. Kinetic energy uses both the velocity and mass to give

So now the question becomes, at what point is this large enough to break glass? Lucky for us, a student at the CUNY College of Criminal Justice wrote their thesis on shooting BBs at windows!

Table 2

This gives the minimum energy for damage as around 2 kJ. We can plot energy vs hail size and see how big the stones have to get:

According to this, the glass will start to be damaged at a diameter of about 140 mm, or the size of a softball! At that point, I think there may be more to worry about than just the skylight.