When a car is at rest,


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

  1. $0.41$

  2. $0.50$

  3. $0.37$

  4. $0.73$

Correct Option: , 4


Rain is falling vertically downwards.

$\overrightarrow{\mathrm{V}}_{\mathrm{r} / \mathrm{m}}=\overrightarrow{\mathrm{V}}_{\mathrm{r}}-\overrightarrow{\mathrm{v}}_{\mathrm{m}}$

$\tan 60^{\circ}=\frac{v_{r}}{v_{m}}=\sqrt{3}$

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

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

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

$\tan 45=\frac{v_{r}}{v_{m}}=1$


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


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

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