The ratio of minimum kinetic energies of two projectiles of same mass is $4: 1$ and the ratio of maximum heights attained by them is $4: 1$. Then the ratio of their ranges is . . . . . . (in $: 1$)

The figure shows the velocity and the acceleration of a point-like body at the initial moment of its motion. The direction and the absolute value of the acceleration remain constant. Find the time when the speed becomes minimum. (Given: $a = 4\, m/s^2$,$v_0 = 40\, m/s$,$\phi = 143^o$)

When a body moves with a constant speed along a circle,

$A$ particle is moving in a uniform circular motion with angular momentum $L$. If the frequency of motion is doubled and its kinetic energy is halved,find the new value of its angular momentum.

$A$ 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 $g = 10 \; m s^{-2}$).

$A$ car runs at a constant speed on a circular track of radius $100\,m$,taking $62.8\,s$ for every circular lap. The average velocity and average speed for each circular lap respectively are: