When a car is at rest, its driver sees rain drops falling on it v… - Vidyadip

Question

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 coming at an angle $60^{\circ}$ from the horizontal. On further increasing the speed of the car to $(1+\beta) v ,$ this angle changes to $45^{\circ} .$ The value of $\beta$ is close to$......$

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Answer

Rain is falling vertically downwards.

$\vec{v}_{r / m}=\vec{v}_{r}-\vec{v}_{m}$

$\tan 60^{\circ}=\frac{ V _{ r }}{ v _{ m }}=\sqrt{3}$

$v _{ r }= v _{ m } \sqrt{3}= v \sqrt{3}$

Now, $v _{ m }=(1+ B ) v$

and $\theta=45^{\circ}$

$\tan 45=\frac{ v _{ r }}{ v _{ m }}=1$

$v _{ r }= v _{ m }$

$v \sqrt{3}=(1+\beta) v$

$\sqrt{3}=1+\beta$

$\Rightarrow \beta=\sqrt{3}-1=0.73$

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