$A$ man on a rectilinearly moving cart,facing the direction of motion,throws a ball straight up with respect to himself.

Two trains,one of length $100 \, m$ and another of length $125 \, m$,are moving in mutually opposite directions along parallel lines. They meet each other,each with a speed of $10 \, m/s$. If their accelerations are $0.3 \, m/s^2$ and $0.2 \, m/s^2$ respectively,then the time they take to pass each other will be ......... $s$.

Two stones are thrown up simultaneously with initial speeds of $u_1$ and $u_2$ $(u_2 > u_1)$. They hit the ground after $6 \, s$ and $10 \, s$ respectively. Which graph in the figure correctly represents the time variation of $\Delta x = (x_2 - x_1)$,the relative position of the second stone with respect to the first up to $t = 10 \, s$? Assume that the stones do not rebound after hitting the ground.

$A$ ball is dropped from a building of height $45 \, m$. Simultaneously,another ball is thrown up with a speed of $40 \, ms^{-1}$. Calculate the relative speed of the balls as a function of time.

$A$ man is $45 \, m$ behind the bus when the bus starts accelerating from rest with an acceleration of $2.5 \, m/s^2$. With what minimum velocity should the man start running to catch the bus?

$A$ police party is moving in a jeep at a constant speed $v$. They see a thief at a distance $x$ on a motorcycle which is at rest. The moment the police see the thief, the thief starts moving with a constant acceleration $a$. Which of the following relations is true if the police are able to catch the thief?