$A$ man standing on a road has to hold his umbrella at $30^{\circ}$ with the vertical to keep the rain away. He throws the umbrella and starts running at $10 \,km/h$. He finds that raindrops are hitting his head vertically. The actual speed of raindrops is:

Two persons $A$ and $B$ are located in the $X-Y$ plane at the points $(0,0)$ and $(0,10)$ respectively. (The distances are measured in $MKS$ units). At time $t=0$, they start moving simultaneously with velocities $\overrightarrow{v}_A = 2\hat{j} \text{ m/s}$ and $\overrightarrow{v}_B = 2\hat{i} \text{ m/s}$ respectively. The time after which $A$ and $B$ are at their closest distance is:

Rain is falling vertically with a speed of $35 \; m s^{-1}$. Winds start blowing after sometime with a speed of $12 \; m s^{-1}$ in the east to west direction. In which direction should a boy waiting at a bus stop hold his umbrella?

$A$ boat is moving with a velocity $3i + 4j$ with respect to the ground. The water in the river is moving with a velocity $-3i - 4j$ with respect to the ground. The relative velocity of the boat with respect to the water is:

Two girls are standing at the ends $A$ and $B$ of a ground where $AB = b$. The girl at $B$ starts running in a direction perpendicular to $AB$ with velocity $V_1$. The girl at $A$ starts running simultaneously with velocity $V_2$ and meets the other girl in the shortest possible time $t$. The value of $t$ is:

$A$ boat crossing a river moves with a velocity $v$ relative to still water. The river is flowing with a velocity $v/2$ with respect to the bank. The angle with respect to the flow direction with which the boat should move to minimize the drift is (in $^{\circ}$)