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. Find the distance (in meters) of the particle from the origin at $t = 2 \, s$.

$Assertion$ : When a particle moves in a circle with a uniform speed,its velocity and acceleration both change.
$Reason$ : The centripetal acceleration in circular motion is dependent on the angular velocity of the body.

The maximum range of a bullet fired from a toy pistol mounted on a car at rest is $R_0 = 10 \, m$. What will be the acute angle of inclination of the pistol for maximum range when the car is moving in the direction of firing with uniform velocity $v = 20 \, m/s$,on a horizontal surface? $(g = 10 \, m/s^2)$

$A$ point $P$ on the circumference of a wheel is initially in contact with the ground. If the radius of the wheel is $5 \, m$,what will be the displacement of point $P$ when the wheel rolls forward by half a revolution?

$A$ particle is projected with velocity $u$ from point $M$ on a frictionless incline of length $20\sqrt{2} \, m$. If it crosses a $40 \, m$ wide well at an angle of $45^o$,what should be its velocity at point $M$?

$A$ projectile of mass $200 \ g$ is launched in a viscous medium at an angle $60^{\circ}$ with the horizontal,with an initial velocity of $270 \ m/s$. It experiences a viscous drag force $\vec{F} = -c \vec{v}$,where the drag coefficient $c = 0.1 \ kg/s$ and $\vec{v}$ is the instantaneous velocity of the projectile. The projectile hits a vertical wall after $2 \ s$. Taking $e = 2.7$,the horizontal distance of the wall from the point of projection (in $m$) is