$A$ bird flies to and fro between two cars which move towards each other with velocities $v_1 = 20\, m/s$ and $v_2 = 30\, m/s$. If the speed of the bird is $v_3 = 10\, m/s$ and the initial distance of separation between them is $d = 2\, km$,find the total distance covered by the bird until the cars meet. (in $, m$)

Two trains $A$ and $B$ of length $400 \; m$ each are moving on two parallel tracks with a uniform speed of $72 \; km \; h^{-1}$ in the same direction,with $A$ ahead of $B$. The driver of $B$ decides to overtake $A$ and accelerates by $1 \; m \; s^{-2}$. If after $50 \; s$,the guard of $B$ just brushes past the driver of $A$,what was the original distance (in $m$) between them?

Two particles are at rest on a straight track and they are separated from each other by $100 \ m$. They start their motion in the same direction,with the same acceleration $10 \ m/s^2$. Their relative displacement after $10 \ s$ will be

$A$ train is running at a constant speed of $90 \,km/h$ on a straight track. $A$ person standing on the top of a bogie moves in the direction of motion of the train such that he covers $1 \,m$ on the train each second. The speed of the person with respect to the ground is .........

$A$ boy sitting on the topmost berth in the compartment of a train which is just going to stop at a railway station,drops an apple aiming at the open hand of his brother sitting vertically below his hands at a distance of about $2 \ m$. The apple will fall:

Two particles start simultaneously from the same point and move along two straight lines,one with uniform velocity $v$ and the other with a uniform acceleration $a$. If $\alpha$ is the angle between the lines of motion of the two particles,then the least value of relative velocity will be at time given by