At $t = 0$,a projectile is fired from a point $O$ (taken as origin) on the ground with a speed of $50 \ m/s$ at an angle of $53^{\circ}$ with the horizontal. It passes through two points $A$ and $B$,each at a height of $75 \ m$ above the horizontal,as shown in the figure. The horizontal separation between the points $A$ and $B$ is . . . . . . $m$.

$A$ person walks in such a way that he covers equal distance in each step. The person takes $2$ steps forward towards East, then takes a right turn and walks $4$ steps towards South, then takes a right turn and walks $6$ steps towards West, and then takes a right turn and walks further. The direction of his final position after a total of $20$ steps walk with respect to his initial position is:

$A$ person is standing on an open car moving with a constant velocity of $30\,m/s$ on a straight horizontal road. The man throws a ball in the vertically upward direction and it returns to the person after the car has moved $240\,m$. Find the speed and the angle of projection.

$A$ particle at a distance of $1 \ m$ from the origin starts moving,such that $dr/d\theta = r$,where $r$ and $\theta$ are polar coordinates. Then,the angle between the resultant velocity and the tangential velocity is

$A$ car is moving at a speed of $72 \, km/hr.$ The diameter of its wheels is $0.5 \, m.$ If the wheels are stopped in $20$ rotations by applying brakes,then the angular retardation produced by the brakes is ............ $rad/s^2$. (in $.5$)

The kinetic energy $K$ of a particle moving along a circle of radius $R$ depends upon the distance $s$ as $K = as^2$. The force acting on the particle is