Question

During a heavy rain, hailstones of average size 1.0 cm in diameter fall with an average speed of 20 m/s. Suppose 2000 hailstones strike every square meter of a 10 m × 10 m roof perpendicularly in one second and assume that the hailstones do not rebound. Calculate the average force exerted by the falling hailstones on the roof. Density of a hailstone is 900 kg/m³.

Answer 1

It is given that:
Diameter of hailstone =  1 cm =  0.01 m
⇒ Radius of hailstone, r =  0.005 m 

Average speed of hailstone =  20 m/s
Density of hailstone =  900 kg/m3 = 0.9 g/cm3
$$\text{ Volume of the hailstones is given as, }$$ 
$$V=\frac{4}{3}\pi r^3$$
$$\Rightarrow V=\frac{4}{3}\pi (0.005{)}^3=5.235\times {10}^{-7}{m}^3$$
$$\text{ Mass = volume }\times \text{ density }=5.235\times {10}^{-7}\times 900$$
$$=4.711\times {10}^{-4}\text{ kg }$$
$$\therefore \text{ Mass of 2000 hailstone }=2000\times 4.711\times {10}^{-4}=0.9422$$
$$\text{ Rate of change of momentum }=0.9422\times 20\approx 19{\text{N/ m}}^2$$

∴ The total force exerted on the roof = 19 × 100 = 1900 N