Car $A$ is moving to the East with a speed of $30 \text{ km/h}$, and car $B$ is moving to the North with the same speed. What is the velocity of car $B$ as measured from car $A$?

$A$ steamboat goes across a lake and comes back $(a)$ on a quiet day when the water is still and $(b)$ on a rough day when there is a uniform air current so as to help the journey onward and to impede the journey back. If the speed of the launch on both days was the same,in which case will it complete the journey in lesser time?

Two particles having position vectors $\overrightarrow{r_1} = (3\hat{i} + 5\hat{j}) \text{ m}$ and $\overrightarrow{r_2} = (-5\hat{i} - 3\hat{j}) \text{ m}$ are moving with velocities $\overrightarrow{v_1} = (4\hat{i} + 3\hat{j}) \text{ m/s}$ and $\overrightarrow{v_2} = (\alpha\hat{i} + 7\hat{j}) \text{ m/s}$. If they collide after $2 \text{ s}$,the value of $\alpha$ is:

On a calm day,a boat can go across a lake and return in time $T_0$ at a speed $V$. On a rough day,there is a uniform current at speed $v$ that helps the onward journey and impedes the return journey. If the time taken to go across and return on the rough day is $T$,then $T / T_0$ is:

$A$ man is crossing a river flowing with a velocity of $5\, m/s$. He reaches a point directly across at a distance of $60\, m$ in $5\, s$. His velocity in still water should be ........ $m/s$.

$A$ man standing on a road holds 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 speed of raindrops with respect to the moving man will be: