$A$ particle of mass $m$ is projected with a speed $u$ from the ground at an angle $\theta = \frac{\pi}{3}$ w.r.t. horizontal ($x$-axis). When it has reached its maximum height,it collides completely inelastically with another particle of the same mass and velocity $u \hat{i}$. The horizontal distance covered by the combined mass before reaching the ground is

Two particles $P$ and $Q$ each of mass $3m$ lie at rest on the $X$-axis at points $(-a, 0)$ and $(+a, 0)$,respectively. $A$ third particle $R$ of mass $2m$ initially at the origin moves towards the particle $Q$ with velocity $v$. If all the collisions of the system of $3$ particles are elastic and head-on,the total number of collisions in the system is

Answer the following:
$(a)$ The casing of a rocket in flight burns up due to friction. At whose expense is the heat energy required for burning obtained? The rocket or the atmosphere?
$(b)$ Comets move around the sun in highly elliptical orbits. The gravitational force on the comet due to the sun is not normal to the comet's velocity in general. Yet the work done by the gravitational force over every complete orbit of the comet is zero. Why?
$(c)$ An artificial satellite orbiting the earth in a very thin atmosphere loses its energy gradually due to dissipation against atmospheric resistance,however small. Why then does its speed increase progressively as it comes closer and closer to the earth?
$(d)$ In Figure $(i)$ the man walks $2\; m$ carrying a mass of $15\; kg$ on his hands. In Figure $(ii)$,he walks the same distance pulling the rope behind him. The rope goes over a pulley,and a mass of $15\; kg$ hangs at its other end. In which case is the work done greater?

$A$ spring-block system is resting on a frictionless floor as shown in the figure. The spring constant is $2.0 \,N \,m^{-1}$ and the mass of the block is $2.0 \,kg$. Ignore the mass of the spring. Initially, the spring is in an unstretched condition. Another block of mass $1.0 \,kg$ moving with a speed of $2.0 \,m \,s^{-1}$ collides elastically with the first block. The collision is such that the $2.0 \,kg$ block does not hit the wall. The distance, in metres, between the two blocks when the spring returns to its unstretched position for the first time after the collision is. . . .

$A$ sphere of mass $m$ slides down a smooth inclined plane from a point $B$ at a height of $h$ starting from rest. The magnitude of the change in momentum of the particle between position $A$ (at the bottom of the incline) and $C$ (on the horizontal surface) is (assuming the angle of inclination of the plane is $\theta$ with respect to the horizontal):

In the diagram shown,there is no friction at any contact surface. Initially,the spring has no deformation. What will be the maximum deformation in the spring? Consider all the strings to be sufficiently large. Consider the spring constant to be $K$.