$A$ particle moves once around a circle of radius $1 \,m$. The time taken is $10 \,s$. The average velocity of the motion is

The displacement vector of a particle of mass $m$ is given by $\vec{r}(t) = A \cos \omega t \hat{i} + B \sin \omega t \hat{j}$.
$(a)$ Show that the trajectory is an ellipse.
$(b)$ Show that $\vec{F} = -m \omega^2 \vec{r}$.

$A$ fighter plane flying horizontally at an altitude of $1.5 \; km$ with a speed of $720 \; km/h$ passes directly overhead an anti-aircraft gun. At what angle from the vertical should the gun be fired for the shell with a muzzle speed of $600 \; m s^{-1}$ to hit the plane? At what minimum altitude should the pilot fly the plane to avoid being hit? (Take $g = 10 \; m s^{-2}$).

$A$ particle of mass $m$ is performing $U.C.M.$ along a circle of radius $r$. The relation between centripetal acceleration $a$ and kinetic energy $E$ is given by

$A$ particle is projected from the ground with an initial speed $v$ at an angle $\theta$ with the horizontal. The average velocity of the particle between its point of projection and the highest point of its trajectory is

$A$ 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: