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:

$A$ particle is moving along the $x$-axis with its coordinate with time '$t$' given by $x(t) = 10 + 8t - 3t^2$. Another particle is moving along the $y$-axis with its coordinate as a function of time given by $y(t) = 5 - 8t^3$. At $t = 1 \; s$,the speed of the second particle as measured in the frame of the first particle is given as $\sqrt{v}$. Then $v$ (in $m^2/s^2$) is:

$A$ ship $X$ moving due north with speed $v$ observes that another ship $Y$ is moving due west with the same speed $v$. The actual velocity of $Y$ is $........$.

$A$ ship $A$ is moving Westwards with a speed of $10 \, km \, h^{-1}$ and a ship $B$,$100 \, km$ South of $A$,is moving Northwards with a speed of $10 \, km \, h^{-1}$. The time after which the distance between them becomes shortest is ........ $hr$.

Two cities $X$ and $Y$ are connected by a regular bus service with a bus leaving in either direction every $T$ min. $A$ girl is driving a scooty with a speed of $60 \ km/h$ in the direction $X$ to $Y$ and notices that a bus goes past her every $30$ minutes in the direction of her motion,and every $10$ minutes in the opposite direction. Choose the correct option for the period $T$ of the bus service and the speed (assumed constant) of the buses.

$A$ shell is fired at a speed of $200\ m/s$ at an angle of $37^o$ above the horizontal from the top of a tower $80\ m$ high. At the same instant,another shell is fired from a jeep travelling away from the tower at a speed of $10\ m/s$ as shown. The velocity of this shell relative to the jeep is $250\ m/s$ at an angle of $53^o$ with the horizontal. Find the time (in $sec$) taken by the two shells to come closest.