$A$ projectile fired at $30^{\circ}$ to the ground is observed to be at the same height at time $t_1 = 3 \, s$ and $t_2 = 5 \, s$ after projection,during its flight. The speed of projection of the projectile is $......... \, m \, s^{-1}$ (Given $g = 10 \, m \, s^{-2}$).

$A$ particle crossing the origin at time $t=0$,moves in the $xy$-plane with a constant acceleration $a$ in the $y$-direction. If the equation of motion of the particle is $y = bx^2$ (where $b$ is a constant),then its velocity component in the $x$-direction is

$A$ particle moves in space according to the equation $\vec{r} = (\sin t \hat{i} + \cos t \hat{j} + t \hat{k}) \text{ m}$. Find the time $t$ when the position vector and acceleration vector are perpendicular to each other.

$A$ particle performs uniform circular motion with an angular momentum $L$. If the frequency of the particle's motion is doubled and its kinetic energy is halved,then its angular momentum becomes

$A$ ball is hit by a batsman at an angle of $37^o$ with an initial velocity $u = 15 \ m/s$ as shown in the figure. The man standing at $P$ is at a distance of $9 \ m$ from the point $B$ where the ball would strike the ground. What is the minimum velocity at which the man should run to catch the ball before it hits the ground? Assume the height of the man is negligible. ........ $m/s$

The velocities of two particles $A$ and $B$,separated by a distance of $20 \ m$,are as shown in the figure. Find the angular velocity of $B$ with respect to $A$ in $rad/s$.