Question

It is a common observation that rain clouds can be at about a kilometer altitude above the grounds.

(a) If a rain drop falls from such a height freely under gravity, what will be its speed? Also calculate in km/h. (g = 10 m/s² )

(b) A typical rain drop is about 4 mm diameter. Momentum is mass × speed in magnitude. Estimate its momentum when it hits ground.

(c) Estimate the time required to flatten the drop.

(d) Rate of change of momentum is force. Estimate how much force such a drop would exert on you.

(e) Estimate the order of magnitude force on umbrella. Typical lateral separation between two rain drops is 5 cm.

(Assume that umbrella is circular and has a diameter of 1 m and cloth is not pierced through !!)

Answer 1

(a) v = \(\sqrt {2gh}\) = \(\sqrt{2\times10\times1000}\) = 141 m/s = 507.6 km/h.

(b) \(m=\frac{4\pi}{3}r^3ρ=\frac{4\pi}{3}{(2\times10^{-3})}^3(10^3)\) = 3.4 x 10^-5 kg.

ρ = mv ≈ 4.7 × 10^-3 kg m/s ≈ 5 × 10^-3 kg m/s.

(c) Diameter ≈ 4 mm

∆t ≈ \(\frac {d}{v}\) = 28 μs ≈ 30 μs

(d) F = \(\frac{\Delta p}{\Delta t}\) = \(\frac{4.7\times 10^{-3}}{28\times10^{-4}}\) ≈ 168 N ≈ 1.7 × 10² N.

(e) Area of cross-section =\(\frac{\pi d^2}{4}\) ≈ 0.8 m².

With average separation of 5 cm, no. of drops that will fall almost simultaneously is 

\(\frac{0.8\, m^2}{{(5\times10{-2})}^2}\) ≈ 320.

Net force ≈ 54000 N

(Practically drops are damped by air viscosity).