The position vector of a particle is given as $\vec{r} = (t^2 - 4t + 6)\hat{i} + (t^2)\hat{j}$. The time after which the velocity vector and acceleration vector become perpendicular to each other is equal to ....... $sec$.

$A$ body of mass $100 \, g$ is projected at an angle of $30^\circ$ with the horizontal with a velocity of $20 \, m \, s^{-1}$. What is the change in its momentum at the maximum height in $kg \, m \, s^{-1}$?

$A$ 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

$A$ stone of mass $m$ is tied to a string of length $l$ and rotated in a circle with a constant speed $v$. If the string is released,the stone flies:

Two particles are projected simultaneously in the same vertical plane,from the same point on the ground,with the same speeds but at different angles $( < 90^o )$ to the horizontal. The path followed by one,as seen by the other,is

Two objects are projected at an angle $\theta^{\circ}$ and $(90-\theta)^{\circ}$ to the horizontal with the same speed. The ratio of their maximum vertical heights is