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
$\sqrt {{u^2} - 2gL} $
$\sqrt {2gL} $
$\sqrt {{u^2} - gL} $
$\sqrt {2\left( {{u^2} - gL} \right)} $
A ball moving with velocity $2\, m/s$ collides head-on with another stationary ball of double the mass. If the coefficient of restitution is $0.5$, then their velocities (in $m/s$) after collision will be
A vertical spring with force constant $k$ is fixed on a table. A ball of mass $m$ at a height $h$ above the free upper end of the spring falls vertically on the spring so that the spring is compressed by a distance $d$. The net work done in the process is
Underline the correct alternative :
$(a)$ When a conservative force does positive work on a body, the potential energy of the body increases/decreases/remains unaltered.
$(b)$ Work done by a body against friction always results in a loss of its kinetic/potential energy.
$(c)$ The rate of change of total momentum of a many-particle system is proportional to the external force/sum of the internal forces on the system.
$(d)$ In an inelastic collision of two bodies, the quantities which do not change after the collision are the total kinetic energy/total linear momentum/total energy of the system of two bodies.
Work done in time $t $ on a body of mass $m$ which is accelerated from rest to a speed $v$ in time ${t_1}$ as a function of time $t$ is given by
A small block of mass $m$ slides along a smooth frictional track as shown in the figure. If it starts from rest at $P$ , velocity of block at point $Q$ is