A particle moves in a straight line with retardation proportional to its displacement. Its loss of kinetic energy for any displacement $x$ is proportional to

A particle moves under the effect of a force $F = cx$ from $x = 0$ to $x = x_1$. The work done in the process is

The potential energy of a long spring when stretched by $2\, cm$ is $U.$ If the spring is stretched by $8\, cm$ the potential energy stored in it is

A shell is fired from a canon with a velocity $V$ at an angle $\theta$ with the horizontal direction. At the highest point in its path, it explodes into two pieces of equal masses. One of the pieces come to rest. The speed of the other piece immediately after the explosion is

A $300\ kg$ crate is dropped vertically onto a conveyor belt that is moving at $1.20\ m/s$ . A motor maintains the belt's constant speed. The belt initially slides under the crate, with a coefficient of friction of $0.400$ . After a short time, the crate is moving at the speed of the belt. During the period in which the crate is being accelerated, find the work done by the motor which drives the belt ................... $\mathrm{J}$

The force acting on a body moving along $x-$ axis varies with the position of the particle as shown in the figure. The body is in stable equilibrium at