A body of mass $2\, kg$ slides down a curved track which is quadrant of a circle of radius $1$ $meter$ as shown in figure. All the surfaces are frictionless. If the body starts from rest, its speed at the bottom of the track is ............. $\mathrm{m}/ \mathrm{s}$

A force $F = - K(yi + xj)$ (where K is a positive constant) acts on a particle moving in the xy-plane. Starting from the origin, the particle is taken along the positive x-axis to the point $(a, 0)$ and then parallel to the y-axis to the point $(a, a)$. The total work done by the force F on the particles is

A body of mass $m$ is projected from ground with speed $u$ at an angle $\theta$ with horizontal. The power delivered by gravity to it at half of maximum height from ground is

Three particles of masses $10g, 20g$ and $40g$ are moving with velocities $10\widehat i,10\widehat j$ and  $10\widehat k$ $m/s$ respectively. If due to some mutual interaction, the first particle comes to  rest and the velocity of second particle becomes $\left( {3\widehat i + 4\widehat j\,\,} \right)\, m/s$, then the velocity of third particle is

A stone tied to a string $L$ is whirled in a vertical circle, with the other end of the string at the centre. At a certain instant of time, the stone is as its lowest position and has a speed $u$. the magnitude of the change in its velocity as it reaches a position where the string is horizontal is

A body of mass $m$ moving with velocity $v$ collides head on with another body of mass  $2\, m$ which is initially at rest. The ratio of $K.E.$ of the colliding body before and  after collision will be