$A$ 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:

$A$ bird is tossing (flying to and fro) between two cars moving towards each other on a straight road. One car has a speed of $54 \text{ kmh}^{-1}$ while the other has a speed of $36 \text{ kmh}^{-1}$. The bird starts moving from the first car towards the other and is moving with a speed of $36 \text{ kmh}^{-1}$ when the two cars are separated by $36 \text{ km}$. The total distance covered by the bird before the cars meet each other is: (in $\text{ m}$)

There are two particles $P$ and $Q$ separated by a distance of $10 \ km$ and moving with velocities of $10 \ kmph$ as shown in the figure. The time elapsed before they reach their minimum separation is.........$hr$.

Two parallel rail tracks run north-south. Train $A$ moves north with a speed of $54 \; km \; h^{-1},$ and train $B$ moves south with a speed of $90 \; km \; h^{-1}.$ What is the velocity of $B$ with respect to $A$?

Two towns $A$ and $B$ are connected by a regular bus service with a bus leaving in either direction every $T$ minutes. $A$ man cycling with a speed of $20 \; km/h$ in the direction $A$ to $B$ notices that a bus goes past him every $18 \; min$ in the direction of his motion,and every $6 \; min$ in the opposite direction. What is the period $T$ (in $min$) of the bus service and with what speed (assumed constant) do the buses ply on the road?

$A$ man is running on a level road with a speed of $10 \ m/s$. He feels that the rain is falling vertically downward with a speed of $10 \sqrt{3} \ m/s$. If he runs in the opposite direction with the same speed,then the speed of the rain relative to him will be: