$A$ stationary man observes that the rain is falling vertically downward. When he starts running with a velocity of $12 \ km/h$,he observes that the rain is falling at an angle of $60^{\circ}$ with the vertical. The actual velocity of the rain is:

An airplane airspeed indicator reads $100 \, m/s$ and its compass shows that it is heading $37^{\circ}$ east of north. The meteorological information provided to the navigator is that the wind velocity is $20 \, m/s$ towards east. The speed of the airplane relative to the ground is closest to ............ $\, m/s$.

$A$ man can swim with a speed of $4.0 \; km/h$ in still water. How long does he take to cross a river $1.0 \; km$ wide if the river flows steadily at $3.0 \; km/h$ and he makes his strokes normal to the river current? How far down the river does he go when he reaches the other bank?

$A$ man of mass $60 \,kg$ is standing on a boat of mass $140 \,kg$,which is at rest in still water. The man is initially at $20 \,m$ from the shore. He starts walking on the boat for $4 \,s$ with a constant speed of $1.5 \,m/s$ towards the shore. The final distance of the man from the shore is .......... $m$.

An observer moves with a constant speed along the line joining two stationary objects. He will observe that the two objects

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: