$A$ 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 $m$)

One end of a rod of length $L=1 \,m$ is fixed to a point on the circumference of a wheel of radius $R=1 / \sqrt{3} \,m$. The other end is sliding freely along a straight channel passing through the centre $O$ of the wheel as shown in the figure. The wheel is rotating with a constant angular velocity $\omega$ about $O$. The speed of the sliding end $P$,when $\theta=60^{\circ}$ is

$A$ 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 $\theta$?

Find the speed of the intersection point $O$ of the two wires if the wires start moving perpendicular to themselves with speed $v$ as shown in the figure.

$A$ projectile is launched from point $A$ of the given landscape with a water body as shown in the diagram. The launching angle is $15^{\circ}$. From the following, identify the right initial velocity of the projectile with which it will fall somewhere in between the points $C$ and $D$. [Assume $g = 10 \,m/s^2$] (in $\,m/s$)

The figure shows a sphere of mass $500 \ g$ moving in a steady flow of air in the $x$-direction. The air-stream exerts an essentially constant force of $0.9 \ N$ on the sphere in the $x$-direction. If at $t = 0$ the sphere is moving as shown in the figure,determine the time required for the sphere to cross the $y$-axis again in $\sec$. (in $.66$)