$A$ passenger sitting in a train $A$ moving at $90\,km/h$ observes another train $B$ moving in the opposite direction for $8\,s$. If the velocity of the train $B$ is $54\,km/h$,then the length of train $B$ is $..........\,m$.

$A$ 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 $10 \ km/hr$. He finds that raindrops are hitting his head vertically. The speed of raindrops with respect to the road will be ......... $km/hr$.

$A$ slider block $A$ moves downward at a speed of $v_A = 2 \ m/s$,at an angle of $75^\circ$ with the horizontal as shown in the figure. The velocity of the portion of belt $B$ between ideal pulleys $C$ and $D$ with respect to $A$ is $v_{CD/A} = 2 \ m/s$ at an angle $\theta$ with the horizontal. The magnitude of the velocity of portion $CD$ of the belt when $\theta = 15^\circ$ is .......... $m/s$.

$A$ 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?

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

Two particles $P_1$ and $P_2$ are moving with velocities $v_1$ and $v_2$ respectively. Which of the following statements about their relative velocity $v_{12}$ is true?