It is a common observation that rain clouds can be at about a kilometer altitude above the ground. (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//s^(2)`).
(b) A typical rain drop is about 4 mm diameter. Momentum is mass x 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.)

Explanation

Given, h = 1 km = 100 m
`g = 10 m //s ^(2)`
(a) Speed of drop in free fall,
`v ^(2) = 2 gh`
`therefore v = sqrt (2gh)`
(b) Diameter of the drop,
`d =4 mm`
`therefore ` Radius of the drop,
`r = 2 mm = 2 xx 10 ^(-3) m`
Mass of a rain drop,
`m = rho xx v `
`=4/3 pi r ^(3) rho = 4/3 xx (22)/(7) xx (2 xx 10 ^(-3)) ^(3) xx 10 ^(3) `
`(because rho = 10 ^(3) kg//m ^(2))`
`~~ 3.4 xx 10 ^(-5) kg`
Momentum of the rain drop,
`= p = mv`
`= 3.4 xx 10 ^(-5) xx 100 sqrt2`
`=4.7 xx 10 ^(-3) kg m//s`
`~~5 xx 10 ^(-3) kgm//s`
(c) Time required to flatten the drop = time taken by the drop to travel the distance equal to the diameter near the ground,
`t = (d)/(v)= (4 xx 10 ^(-3))/(1 00 sqrt2) = 0.028 xx 10^(-3) s`
`=2.8 xx 10 ^(-5) s ~~ 30 ms`
(d) Force exerted by a rain drop,
`F = ("Change in momentum")/("Time") = (Delta p )/(t) = (p -0)/(t) = (4.7 xx 10 ^(-3))/(2.8 xx 10 ^(-5)) ~168 N`
(e) Radius of the unbrella `=R = 1/2 m`
`therefore `Area of the unbrella,
`A = pi R ^(2) = (22)/(7) xx ((1)/(2)) ^(2) = (22)/(28) = (11)/(14) =0.8 m ^(2) `
Number of drops striking the umbrella,
`= ("Area of umbrella")/("seperation")= (0.8)/((5 xx 10 ^(-2))^(2)) = 320`
`therefore ` Net force `= 320 xx 168 = 53760 N = 54000 N`