$A$ $2 \ kg$ ball is thrown vertically upward and another $3 \ kg$ ball is projected with a certain angle $(\theta \neq 90^{\circ})$. Both have the same time of flight. The ratio of their maximum heights is:

Two bodies are projected from points $(0,0)$ and $(\sqrt{3}-1,0)$ with velocities $10 \ ms^{-1}$ and $v \ ms^{-1}$ respectively,as shown in the figure. The time after which they collide in space is . . . . . . . (in $s$)

$A$ particle is moving in a circle of radius $R$ with constant speed $V$. The magnitude of average acceleration after half revolution is

$A$ particle moves in the $XY$-plane with $x$ and $y$ coordinates varying with time $t$ as $x(t) = 5t$ and $y(t) = 5t(27 - t^2)$. At what time $t$ (in seconds) will the direction of velocity and acceleration be perpendicular to each other?

$A$ body is moving with velocity $30 \; m/s$ towards east. After $10 \; s$,its velocity becomes $40 \; m/s$ towards north. The average acceleration of the body is ...... $m/s^2$.

$A$ train is moving along a straight line with a constant acceleration '$a$'. $A$ boy standing in the train throws a ball forward with a speed of $10 \ m/s$,at an angle of $60^{\circ}$ to the horizontal. The boy has to move forward by $1.15 \ m$ inside the train to catch the ball back at the initial height. The acceleration of the train,in $m/s^2$,is