A motor car of width **w** moves uniformly along a straight road , parallel to the pavement almost touching it. A pedestrian on the edge of the pavement at a distance **l** ahead of the car begins to walk uniformly to cross the road . If v is the speed of the car and u is the speed of the pedestrian relative to the road , show that the pedestrian can cross the road safely in front of the car

if u > v sin (alpha) , where (alpha) = tan^-1 (w/l).

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Let B be the position of the pedestrian on edg of pavement. A is position of car on the road (consider car as a point on pavement ). Given AB=I unit. Let B' is point opposite to point B on the therside of the road.Let pedestrian crosses road by making an angle `pi/2-alpha` to ine joining BB' perpendicular to pavement and reaches at ponit C.Thus

`BC=Wsec(pi/2-alpha)=Wcosec(alpha)`

`B'C=Wtan(pi/2-alpha)=Wcot(alpha)`

Thus car has to travell dstance =`I+W cot(alpha)`

pedestrian has to travel=`Wcosec(alpha)`

Pedestrian just cross road infront of the car then

`(I+Wcot(alpha))/v>W(cosec(alpha))/u`

`(Isin(alpha)+Wcos(alpha))u>Wv`

`uIsin(alpha)+W(ucos(alpha)-v)>0`

`=> ucos(alpha)-v>0`

`u>vsec(alpha)`

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