Hailstorm Impact Physics - 2D Side View, Terminal Speed and Impact Energy

Presets

Storm

Mean stone diameter

Hail intensity (stones per m2 per s)

Horizontal wind

Drag coefficient Cd (rough sphere)

Panel dent threshold energy

Time speed

Live readouts

Graphs

Terminal speed (m/s) vs stone diameter (mm)

Impact kinetic energy (J, log) vs diameter, with the dent threshold

Size spectrum of the stones that fell (count per 5 mm bin)

Numerical checks

running...

How to use

Stones fall through the side view and strike a roof panel. Their diameters follow an exponential spread around the median you choose. Raise the diameter and the impact energy (colour of the splash: yellow below the dent threshold, red above) grows roughly as the fourth power of the diameter. Add wind to slant the fall, change Cd to see how rough or smooth stones change the terminal speed, and lower the panel threshold to simulate a weaker surface.

The science

Terminal speed. A falling stone of diameter D and ice density rho_i feels weight m g and quadratic drag (1/2) rho_air Cd A v^2, with m = (pi/6) rho_i D^3 and A = (pi/4) D^2. Balancing the two gives

v_t = sqrt( 4 rho_i g D / (3 rho_air Cd) ) , v(t) = v_t tanh(g t / v_t)

so v_t grows as sqrt(D), the mass as D^3, and the impact energy as

E = (1/2) m v_t^2 = (pi/6) rho_i D^3 * (2 rho_i g D / (3 rho_air Cd)) = (pi rho_i^2 g / (9 rho_air Cd)) D^4

Doubling the diameter therefore multiplies the impact energy by 16.

Size spectrum. Hail sizes are modelled with a Marshall-Palmer exponential distribution N(D) = N0 exp(-L D) with mean diameter 1/L (here the slider value). Real distributions depend on the storm.

Honest simplifications. Stones are smooth spheres with a constant Cd (real hail is lumpy and tumbles; Cd depends on Reynolds number, about 0.4 to 0.8), air density is the sea-level value, and wind only slants the path. The dent threshold is an energy that you choose; it is not a calibrated damage law for any material. Only inanimate surfaces are modelled. This is the 2D side-view companion of the 3D hailstorm impact simulator, whose speed is a simple fit; here it is derived from the force balance.