$A$ ball is projected from a point with a speed $V_0$ at a certain angle $\theta$ with the horizontal. From the same point and at the same instant,a person starts running with a constant speed $0.5 V_0$ to catch the ball. If the person catches the ball after some time,then the angle of projection of the ball is (in $^{\circ}$)

Two particles of equal masses are revolving in circular paths of radii $r_1$ and $r_2$ respectively with the same speed. The ratio of their centripetal forces is

Two particles $A$ and $B$ start at the origin $O$ and travel in opposite directions along a circular path of radius $0.5\,m$ at constant speeds $0.5\,m/s$ and $1.5\,m/s$,respectively. The time when they collide with each other is ........ $\text{s}$.

$A$ boy throws a ball with a velocity $V_0$ at an angle $\alpha$ to the ground. At the same time,he starts running with a uniform velocity to catch the ball before it hits the ground. To achieve this,he should run with a velocity of

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.

$A$ strap is passing over a wheel of radius $30\, cm.$ During the time the wheel moving with an initial constant velocity of $2\, rev/s$ comes to rest,the strap covers a distance of $25\, m.$ The deceleration of the wheel in $rad/s^2$ is