Two bodies of masses $m$ and $3m$ are rotating in horizontal circles of radii $r$ and $\frac{r}{3}$ respectively. The tangential speed of the body of mass $m$ is $n$ times that of the heavier body. If the centripetal force is the same for both,the value of $n$ 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 ........ $\text{s}$.

Two projectiles $A$ and $B$ are thrown with the same speed but at angles of $40^{\circ}$ and $50^{\circ}$ with the horizontal,respectively. Then:

What determines the nature of the path followed by a particle?

$A$ particle is projected vertically upwards from $O$ with velocity $v$ and a second particle is projected at the same instant from $P$ (at a height $h$ above $O$) with velocity $v$ at an angle of projection $\theta$. The time when the distance between them is minimum is

$A$ particle is projected from the ground with some initial velocity making an angle of $45^{\circ}$ with the horizontal. If it reaches a height of $7.5 \ m$ above the ground, while it travels a horizontal distance of $10 \ m$ from the point of projection, then the initial speed of the particle is (assume, $g=10 \ m/s^2$): (in $m/s$)