When work is done on a body by an external force,its

$A$ body of mass $m$ starts moving from rest along the $x-$axis such that its velocity varies as $v = a\sqrt{s}$,where $a$ is a constant and $s$ is the distance covered by the body. The total work done by all the forces acting on the body in the first $t$ seconds after the start of the motion is:

The velocity at which the $6 \text{ kg}$ mass (shown in the figure) strikes the ground when it is released from a height of $6 \text{ m}$ above the ground is . . . . . . $\text{m/s}$. Assume the pulley is massless and the string is light and inextensible. (Take $g = 10 \text{ m/s}^2$)

$A$ small block slides down from rest at point $A$ on the surface of a smooth cylinder,as shown. At point $B$,the block falls off (leaves) the cylinder. The equation relating the angles $\theta_1$ and $\theta_2$ is given by

The pulley and spring are massless,and friction is absent everywhere. $A$ $5\, kg$ block is released from rest. Find the speed of the $5\, kg$ block when the $2\, kg$ block leaves contact with the ground. (Take the force constant of the spring $K = 40\, N/m$ and $g = 10\, m/s^2$.)

Three different objects of masses $m_1, m_2$ and $m_3$ are allowed to fall from rest and from the same point $O$ along three different frictionless paths. The speeds of the three objects,on reaching the ground,will be in the ratio of