If the radius of the circular path and the frequency of revolution of a particle of mass $m$ are doubled,then the change in its kinetic energy will be ($E_i$ and $E_f$ are the initial and final kinetic energies of the particle respectively). (in $E_i$)

$A$ 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 is moving with a constant speed of $\sqrt{2} \, m/s$ on a circular path of radius $10 \, cm$. Find the magnitude of the average velocity when it has covered $\frac{3}{4}$ of the circular path.

$A$ particle of mass $m$ is projected at a velocity $u$,making an angle $\theta$ with the horizontal ($x$-axis). If the angle of projection $\theta$ is varied keeping all other parameters same,then the magnitude of angular momentum $(L)$ at its maximum height about the point of projection varies with $\theta$ as,

$A$ particle has initial velocity $(2\hat{i} + 3\hat{j})$ and acceleration $(0.3\hat{i} + 0.2\hat{j})$. The magnitude of velocity after $10\, s$ will be:

At $t = 0$,a projectile is fired from a point $O$ (taken as origin) on the ground with a speed of $50 \ m/s$ at an angle of $53^{\circ}$ with the horizontal. It passes through two points $A$ and $B$,each at a height of $75 \ m$ above the horizontal,as shown in the figure. The horizontal separation between the points $A$ and $B$ is . . . . . . $m$.