In the figure shown,the two projectiles are fired simultaneously. The minimum distance between them during their flight is ........ $m$.

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.

$A$ ring of mass $m$ moves from point $1$ to point $2$ along a smooth rigid horizontal wire with a constant speed $v$. The average force acting on the ring over the time of its motion from $1$ to $2$ is

$A$ projectile is projected at an angle of $30^{\circ}$ from the horizontal with an initial velocity $u$. If the range of the projectile is $R$,what will be the range if the projectile is projected at an angle of $60^{\circ}$ with the same initial velocity?

$A$ particle of mass $1 \ kg$ is projected with an initial velocity $10 \ ms^{-1}$ at an angle of projection $45^{\circ}$ with the horizontal. The average torque acting on the projectile,between the time at which it is projected and the time at which it strikes the ground,about the point of projection in $Nm$ is:

$A$ ball is thrown horizontally from a height with a certain initial velocity at time $t=0$. The ball bounces repeatedly from the ground with the coefficient of restitution less than $1$ as shown below. Neglecting air resistance and taking the upward direction as positive,which figure qualitatively depicts the vertical component of the ball's velocity $v_y$ as a function of time $t$?