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:

To a person,going eastward in a car with a velocity of $25\, km/hr$,a train appears to move towards north with a velocity of $25\sqrt{3}\, km/hr$. The actual velocity of the train will be..........$km/hr$

$A$ man walking briskly in rain with speed $v$ must slant his umbrella forward making an angle $\theta$ with the vertical. $A$ student derives the following relation between $\theta$ and $v: \tan \theta = v$ and checks that the relation has a correct limit: as $v \rightarrow 0, \theta \rightarrow 0,$ as expected. (We are assuming there is no strong wind and that the rain falls vertically for a stationary man). Do you think this relation can be correct? If not,guess the correct relation.

Ram moves in the east direction at a speed of $6 \, m/s$ and Shyam moves $30^{\circ}$ east of north at a speed of $6 \, m/s$. The magnitude of their relative velocity is ........ $m/s$.

$A$ boy standing on the footpath tosses a ball straight up and catches it. The driver of a car passing by,moving with uniform velocity,sees this. The trajectory of the ball as seen by the driver will be:

Two cars get closer by $8\,m$ every second while travelling in the opposite directions. They get closer by $0.8\,m$ in every second while travelling in the same direction. What are the speeds of the two cars?