$A$ particle of mass $m$ is moving in a circular path of constant radius $r$ such that its centripetal acceleration $a$ varies with time $t$ as $a = k^{2} r t^{2}$,where $k$ is a constant. The power delivered to the particle by the force acting on it is given as:

$A$ particle is in uniform circular motion. Its velocity is perpendicular to which of the following?

An object of mass $m$ is projected with a momentum $p$ at such an angle that its maximum height is $\frac{1}{4}$th of its horizontal range. Its minimum kinetic energy in its path will be

$A$ horizontal plane supports a stationary vertical cylinder of radius $R = 1 \ m$ and a disc $A$ attached to the cylinder by a horizontal thread $AB$ of length $l_0 = 2 \ m$ (seen in figure,top view). An initial velocity $v_0 = 1 \ m/s$ is imparted to the disc as shown in the figure. How many seconds will it move along the plane until it strikes against the cylinder? (All surfaces are assumed to be smooth.)

$A$ sportsman runs around a circular track of radius $r$ such that he traverses the path $ABAB$. The distance travelled and displacement,respectively,are

Show that for a projectile,the angle between the velocity and the $x$-axis as a function of time is given by $\theta(t) = \tan^{-1}\left(\frac{v_{0y} - gt}{v_{0x}}\right)$.
Show that the projection angle $\theta_0$ for a projectile launched from the origin is given by $\theta_0 = \tan^{-1}\left(\frac{4h_m}{R}\right)$,where the symbols have their usual meanings.