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 calcualte in `km//h (g = 10 m//s^(2))`.
(b) A typical rain drop is about 4 mm diameter. Momentum is mass `xx` speed in magnitude. Estiamate its momentum when its hits ground.
(c) Estimate the time required to flatten the drop.
(d) Rate of change of momentum is force. Estiamate 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 drop is 5 cm.
(Assume that umbrealla is circular and has a diameter of 1 m and cloth is not peicreced through.)

Explanation

Here, ` h= 1000 m, g=10 m//s^2`
(a) Velcoity of rain drop after fallin height (h) si
` v = sqrt (2 gh) = sqrt (2 xx 10 xx 1000) = 100 sqrt 2 m//s = ( 100 sqer 2 xx 60 xx 60)/(1000) km// h = 36 0 sqrt 2~~ 510 km //h`
(b) Diameter of drop, ` d = 2 =4 mm or r= 2 mm = 2 xxx 10^(-3)m`
Mass of rain drop, ` m= 4/3/ pi r^3 roh = 4/3 xx (220/7 xx ( 2 xx 10^3)^3 xx 10^(-3)^3 xx 10^3 ~~ 3.4 xx 10 ^(-5) kg`
Mumentum of rain drop when it hits the ground, ` P = mu = ( 3.4 xx 10 ^(-5) xx 100 sqrt 2 ~~ 4.7 xx 10^(-3) kg m//s`
(c ) Time required to flatten the frop is
` Delta t = d/v = (4 xx 10^(-3))/(100 sqrt 2) = 0/028 xx 10^(-3) = 28 xx 10^(-6) s`
(d) Force, ` F= (change in momentum )/(time) = ( P-0)/(Delta t) = ( 4. 7 xx 10^(-3))/(28 xx 10^(-6)) ~~ 16 8 N`
(e) Given , rakies of umbrella, ` R = 1//2 m`
Area of umbrella ` =pi R^2 = 922) /7 9 1/2 )^2 = (22)/(28) = (11)/(14) ~~ 0.8 m^@`
with average separaion of ` 5 cm (= 5 xx 10^(-2) m), number of drops those will hit nubrella simultanceously
` = (0.8)/((5 xx 10^(-2))^2`
Net force on umbrella`= 320 xx 168 = 53760 N ~~ 54000 N`.