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Q.

When a car is at rest, its driver sees rain drops falling on it vertically. When driving the car with speed $v$ , he sees that rain drops are falling at an angle 60° from the horizontal. On further increasing the speed of the car to $(1 + β) v,$ this angle changes to 45°. The value of $β$ is close to :

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a

0.37

b

0.50

c

0.41

d

0.73

answer is B.

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Detailed Solution

$\begin{array}{l} {\vec{V}}_{R C} = {\vec{V}}_{R} - {\vec{V}}_{C} \\ S o \tan 30 ° = \frac{ϑ_{C}}{ϑ_{R}} = \frac{1}{\sqrt{3}} \\ \Rightarrow ϑ_{R} = \sqrt{3} ϑ \end{array}$

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Next, let speed of the car is $ϑ^{1} = (1 + β) ϑ$

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$\begin{array}{l} S o, \tan 45^{0} = \frac{ϑ^{1}}{ϑ_{R}} \\ \Rightarrow ϑ^{1} = ϑ_{R} \\ \Rightarrow (1 + β) ϑ = \sqrt{3} ϑ \\ \Rightarrow β = \sqrt{3} - 1 = 0.73 \end{array}$

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