A man running on a horizontal road at 8 km/hr finds the rain

falling vertically. He increase the speed to 12 km/hr and finds

that the drops make angle 30® with the vertical. Find the speed

and the direction ofthe rain with respect to the road.

Asked by jatv4550 | 20th Feb, 2021, 11:40: PM

Expert Answer:

Let the velocity of rain vr = ( x begin mathsize 14px style i with hat on top end style - y begin mathsize 14px style j with hat on top end style ) km/hr  .........................(1)
if velocity of man vm = 8 begin mathsize 14px style i with hat on top end style km/hr , then relative velocity of rain  is given as
( vr - vm ) =  [ ( x - 8 ) begin mathsize 14px style i with hat on top end style - y begin mathsize 14px style j with hat on top end style ] km/hr  ........................ ( 2 )
when man is running at 8 begin mathsize 14px style i with hat on top end style km/hr , rain appears falling vertical , hence in eqn.(1) ,  (x-8) = 0 or x = 8 km/hr
Let us rewrite eqn.(1) by substituting x = 8 km/hr , vr = ( 8 begin mathsize 14px style i with hat on top end style - y begin mathsize 14px style j with hat on top end style ) km/hr  .........................(3)
if velocity of man vm = 12 begin mathsize 14px style i with hat on top end style km/hr , then relative velocity of rain  is given as
( vr - vm ) =  [ ( 8 - 12 ) begin mathsize 14px style i with hat on top end style - y begin mathsize 14px style j with hat on top end style ] km/hr  = -4 begin mathsize 14px style i with hat on top end style - y begin mathsize 14px style j with hat on top end style  ........................ ( 4 )
when man is running at 12 begin mathsize 14px style i with hat on top end style km/hr , rain appears falling at angle 30o with vertical ,
hence from eqn.(4) , y/4 = tan60 (see figure above)  or   y = 4 √3 = 6.93 km/hr
 
Hence velocity of rain , vr = ( 8 begin mathsize 14px style i with hat on top end style - 6.93 begin mathsize 14px style j with hat on top end style ) km/hr
magnitude of velocity of rain , | vr | = { 82 + 6.93 × 6.93 }1/2 = 10.58 km/hr
 
direction of rain makes angle θ with vertical that is given as , θ = tan ( 6.93 / 8 ) = 41o as shown in figure below
 
 

Answered by Thiyagarajan K | 21st Feb, 2021, 08:58: PM

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