A man standing on a road has to hold his umbrella at 3 0 ∘ with the vertical to keep the rain away. He thrown the umbrella and starts running at 10 km / h . He finds that rain drop are hitting his hea

Question

A man standing on a road has to hold his umbrella at 3 0 ∘ with the vertical to keep the rain away. He thrown the umbrella and starts running at 10 km / h . He finds that rain drop are hitting his head vertically. Find the speed of rain w.r.t. road-

TestHub · Accepted Answer

To refine the solution, we can use vector addition.

Let $v_{r}$ be the velocity of rain, $v_{m}$ be the velocity of the man, and $v_{r, m}$ be the velocity of rain relative to the man.

The problem states that when the man is standing, he holds his umbrella at $3 0^{\circ}$ with the vertical. This means the rain's velocity relative to the ground ( $v_{r}$ ) makes an angle of $3 0^{\circ}$ with the vertical.

Let $v_{r} = v_{x} \hat{i} - v_{y} \hat{j}$ .

When standing, $v_{m} = 0$ . So $v_{r, m} = v_{r}$ .

The angle with the vertical is $3 0^{\circ}$ , so $tan 3 0^{\circ} = \frac{v _{x}}{v _{y}} = \frac{1}{3}$ .

Thus, $v_{y} = 3 v_{x}$ .

When the man runs at $10 km / h$ , $v_{m} = 10 \hat{i} km / h$ .

The rain hits his head vertically, meaning $v_{r, m}$ is purely vertical.

$v_{r, m} = v_{r} - v_{m} = (v_{x} \hat{i} - v_{y} \hat{j}) - (10 \hat{i})$

$v_{r, m} = (v_{x} - 10) \hat{i} - v_{y} \hat{j}$

Since $v_{r, m}$ is vertical, its horizontal component must be zero.

So, $v_{x} - 10 = 0 \Rightarrow v_{x} = 10 km / h$ .

Now, we can find $v_{y}$ :

$v_{y} = 3 v_{x} = 3 (10) = 103 km / h$ .

The speed of rain with respect to the road is $∣ v_{r} ∣ = v_{x}^{2} + v_{y}^{2}$ .

$∣ v_{r} ∣ = (10)^{2} + (103)^{2} = 100 + 100 \times 3 = 100 + 300 = 400 = 20 km / h$ .

The final answer is $20 km / h$ .

Physics - Kinematics Question with Solution | TestHub

Options:

Doubts & Discussion