$A$ train is moving at a constant speed $V$. When its driver observes another train in front of him on the same track and moving in the same direction with constant speed $u$. If the distance between the trains is $x$,what should be the minimum retardation of the train to avoid a collision?

$A$ body is projected vertically upward with speed $10 \, m/s$ and another at the same time with the same speed in the downward direction from the top of a tower. The magnitude of the acceleration of the first body with respect to the second is ...... $m/s^2$ (take $g = 10 \, m/s^2$).

$A$ 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:

$A$ $120\, m$ long train $A$ is moving with a speed of $20\, m/s$. $A$ train $B$ of length $130\, m$ is moving with a speed of $30\, m/s$ in the opposite direction. Calculate the time taken by the trains to cross each other. (in $, s$)

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

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?