$A$ projectile of mass $m$ is projected from the ground with a speed of $50 \, m/s$ at an angle of $53^{\circ}$ with the horizontal. It breaks up into two equal parts at the highest point of the trajectory. One particle comes to rest immediately after the explosion. The ratio of the radii of curvatures of the moving particle just before and just after the explosion is:

$A$ particle is moving in the $xy$-plane in a circular path with its center at the origin. If at an instant the position of the particle is given by $\frac{1}{\sqrt{2}}(\hat{i}+\hat{j})$,then the velocity of the particle is along .......

$Assertion$ : 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.

$A$ car is going round a circle of radius $R_1$ with constant speed. Another car is going round a circle of radius $R_2$ with constant speed. If both of them take the same time to complete the circles,the ratio of their angular speeds and linear speeds will be .........

$A$ car moving with a linear speed of $72 \ km/h$ has wheels of radius $0.250 \ m$. If the wheels come to rest after $20$ rotations upon applying brakes,the angular retardation produced by the brakes is ....... $rad \ s^{-2}$. (in $.5$)

$A$ frictionless circular wire of unit radius is fixed on the horizontal plane. Two point particles of unit mass start moving simultaneously from point $A(\theta = \pi/2)$ with identical uniform angular speeds in opposite directions, and meet again at point $B(\theta = -\pi/2)$. During this time, which of the following figures schematically represent the magnitude of the total linear momentum $P$ of the system, as a function of $\theta$?