An object is projected with kinetic energy $K$ from a point $A$ at an angle $60^{\circ}$ with the horizontal. In the absence of air friction, what is the ratio of the difference in kinetic energies at points $B$ and $C$ to the kinetic energy at point $A$ (see figure)?

$A$ body of mass $m$ is performing a $UCM$ in a circle of radius $r$ with speed $v$. The work done by the centripetal force in moving it through $\left(\frac{2}{3}\right)$ rd of the circular path is

$A$ body projected along a frictionless inclined plane of angle of inclination $30^{\circ}$ covers a distance of $40\, m$ along the plane. If the body is projected with the same speed at an angle of $30^{\circ}$ with the ground,it will have a range of: (Take $g = 10\, m/s^2$)

The relative angular speed of the hour hand and the minute hand of a clock is:

Two cars $S_1$ and $S_2$ are moving in coplanar concentric circular tracks in the opposite sense with the periods of revolution $3 \, min$ and $24 \, min$,respectively. At time $t = 0$,the cars are farthest apart. Then,the two cars will be

One end of a rod of length $L=1 \,m$ is fixed to a point on the circumference of a wheel of radius $R=1 / \sqrt{3} \,m$. The other end is sliding freely along a straight channel passing through the centre $O$ of the wheel as shown in the figure. The wheel is rotating with a constant angular velocity $\omega$ about $O$. The speed of the sliding end $P$,when $\theta=60^{\circ}$ is