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Question

To a man running at the rate of 4 km/h, the rain appears to fall vertically. When he increases his speed to 8 km/h, rain appears coming to man at an angle of 45 with vertical. Find the magnitude of velocity of rain with respect to ground.

A
32 km/h
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B
42 km/h
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C
22 km/h
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D
52 km/h
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Solution

The correct option is B 42 km/h
Let the velocity of the rain w.r.t ground be vrg=a^i+b^j km/h
where ^i and ^j are the unit vector in horizontal and vertical direction respectively.

Magnitude of velocity of rain w.r.t ground |vrg|=a2+b2 km/h...(1)

In first case,
Velocity of man w.r.t ground vmg=4^i km/h
vrm=vrgvmg=(a4)^i+b^j
Rain appear to man to be in vertical direction, xcomponent of vmg should be zero.
(a4)=0 a=4
In second case;
vmg=8 ^i km/h
vrm=(a8)^i+b^j (but a = 4)
vrm=4^i+b^j

In this case, rain appears to be at 45o with vertical.
tan45=b4b=4
|b|=4
Magnitude of velocity of rain w.r.t ground
|vmg|=a2+b2
|vmg|=42+42=42 km/h

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