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Question

Rain appears to fall vertically downwards to a man walking at the rate of 3 km/h. When he increases his speed to 6 km/h, the rain appears coming to him at an angle of 45 with the vertical. The speed of rain is

A
33 km/h
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B
32 km/h
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C
35 km/h
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D
53 km/h
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Solution

The correct option is B 32 km/h
Let the velocity of rain with respect to ground be, Vr=(x^iy^j) km/h
Case I: Velocity of man w.r.t ground is,Vm=3^i km/h
Velocity of rain w.r.t man is given by,
Vrm=VrVm
Vrm=(x3)^iy^j ...(1)


As rain appears to fall vertically downwards,
θ=90 with Xaxis
tanθ=
From Eq.(1)
yx3=
i.e x3=0
x=3 ...(2)

Case II: When velocity of man becomes, Vm=6^i km/h
Vrm=VrVm
=(x^iy^j)6^i=(x6)^iy^j
Substituting value of x gives,
Vrm=3^iy^j
As rain appears to make 45 with vertical,


tan45=|y||3|=y3
1=y3
y=3
Therefore, velocity of rain w.r.t ground is
Vr=3^i3^j km/h
Now, speed of rain or its magnitude is,
Vr=(3)2+(3)2
Vr=32 km/h
Speed of the rain with respect to ground is 32 km/h.

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