$A$ lift of mass $2000 \ kg$ starts from rest in the basement and moves to the fourth floor,which is at a height of $25 \ m$. As it passes the fourth floor,its speed is $3 \ ms^{-1}$. There is a constant frictional force of $500 \ N$ acting on it. Calculate the work done by the lift's motor in $kJ$.

Two rectangular blocks $A$ and $B$ of masses $2 \,kg$ and $3 \,kg$ respectively are connected by a spring of spring constant $10.8 \,Nm^{-1}$ and are placed on a frictionless horizontal surface. The block $A$ is given an initial velocity of $0.15 \,ms^{-1}$ in the direction shown in the figure. The maximum compression of the spring during the motion is (in $\,m$)

$A$ body of mass $2 \, \text{kg}$ moving with a speed of $4 \, \text{m/s}$ makes an elastic collision with another body at rest and continues to move in the original direction but with one-fourth of its initial speed. The speed of the two-body centre of mass is $\frac{x}{10} \, \text{m/s}$. Then the value of $x$ is ..... .

$A$ particle of unit mass is moving along the $x$-axis under the influence of a force and its total energy is conserved. Four possible forms of the potential energy of the particle are given in column $I$ ($a$ and $U_0$ are constants). Match the potential energies in column $I$ to the corresponding statement$(s)$ in column $II$.

Column $I$	Column $II$
$(A) U_1(x) = \frac{U_0}{2} \left[1 - \left(\frac{x}{a}\right)^2\right]^2$	$(P)$ The force acting on the particle is zero at $x = a$.
$(B) U_2(x) = \frac{U_0}{2} \left(\frac{x}{a}\right)^2$	$(Q)$ The force acting on the particle is zero at $x = 0$.
$(C) U_3(x) = \frac{U_0}{2} \left(\frac{x}{a}\right)^2 \exp \left[-\left(\frac{x}{a}\right)^2\right]$	$(R)$ The force acting on the particle is zero at $x = -a$.
$(D) U_4(x) = \frac{U_0}{2} \left[\frac{x}{a} - \frac{1}{3}\left(\frac{x}{a}\right)^3\right]$	$(S)$ The particle experiences an attractive force towards $x = 0$ in the region $|x| < a$.
	$(T)$ The particle with total energy $\frac{U_0}{4}$ can oscillate about the point $x = -a$.

In an inelastic collision,

Two blocks $A$ and $B$,each of mass $m$,are connected by a massless spring of natural length $L$ and spring constant $K$. The blocks are initially resting on a smooth horizontal floor with the spring at its natural length as shown in the figure. $A$ third identical block $C$,also of mass $m$,moves on the floor with a speed $v$ along the line joining $A$ and $B$ and collides with $A$. Then: