$A$ car goes around a uniform circular track of radius $R$ at a uniform speed $v$ once in every $T$ seconds. The magnitude of the centripetal acceleration is $a_c$. If the car now goes uniformly around a larger circular track of radius $2R$ and experiences a centripetal acceleration of magnitude $8a_c$,then its time period is:

$A$ 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

Particles $A$ and $B$ are moving with constant velocities along the $x$ and $y$ axes respectively. The graph of the separation $S$ between them with time $t$ is:

Two particles are projected from a tower in opposite directions horizontally with speed $10\,m/s$ each. At $t=1\,s$,match the following two columns.

Column $I$	Column $II$
$(A)$ Relative acceleration between two	$(p)$ $0$ $SI$ unit
$(B)$ Relative velocity between two	$(q)$ $5$ $SI$ unit
$(C)$ Horizontal distance between two	$(r)$ $10$ $SI$ unit
$(D)$ Vertical distance between two	$(s)$ $20$ $SI$ unit

$A$ body is projected at $t=0$ with a velocity $10 \ m/s$ at an angle of $60^{\circ}$ with the horizontal. The radius of curvature of its trajectory at $t=1 \ s$ is $R$. Neglecting air resistance and taking acceleration due to gravity $g=10 \ m/s^2$,the value of $R$ is: (in $m$)

$A$ truck is moving on a horizontal road with a constant speed $v$. $A$ ball is thrown from the truck vertically upward at a speed $u$ with respect to the truck. What is the distance traversed by the truck when the ball returns to the truck?