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Solve the following :

Question:

During a heavy rain, hailstones of average size $1.0 \mathrm{~cm}$ in diameter fall with an average speed of $20 \mathrm{~m} / \mathrm{s}$. Suppose 2000 hailstorms strike every square meter of a $10 \mathrm{~m}^{\times} 10 \mathrm{~m}$ roof perpendicularly in one second and the average force exerted by the falling hailstones on the roof. Density of a hailstone is $900^{\mathrm{kg} / \mathrm{m}^{3}}$

Solution:

Volume of 1 hailstorm

$=\frac{4}{3} \pi\left(\frac{10^{-2}}{2}\right)^{3}=\frac{4}{24} \pi \times 10^{-6}$

$=\frac{\pi}{6} \times 10^{-6} \mathrm{~m}^{3}$

Mass $=\rho . V=9000 \times \frac{\pi}{6} \times 10^{-6}$

$=150 \pi \times 10^{-6} \mathrm{~kg}$

Mass of 2000 hailstorms $=2000 \times 150 \pi \times 10^{-6}$

$=0.3 \pi_{\mathrm{kg}}$

average force on $1^{m^{3}}$ roof

$=\frac{\Delta p}{\Delta t} \frac{d p}{d t}=\frac{0.3 \pi \times 20}{1}=18.84 \sim 19$

Average force on $10 \mathrm{~m} \times 10 \mathrm{~m}\left(100^{\mathrm{m}^{2}}\right)$ roof

$=19 \times 100=1900 \mathrm{~N}$