$A$ bomb is kept stationary at a point. It suddenly explodes into two fragments of masses $1\,g$ and $3\,g$. The total kinetic energy ($K$.$E$.) of the fragments is $6.4 \times 10^4\,J$. What is the $K$.$E$. of the smaller fragment?

$A$ simple pendulum of length $L = \frac{10}{3} \text{ m}$ with a bob of mass $M = 3m$ is hanging freely from a rigid support. $A$ bullet of mass $m$ is fired with a velocity $u = 50 \text{ ms}^{-1}$ from the ground at an angle $\theta$ with the horizontal. When the bullet is at its highest point of its trajectory,it collides head-on with the bob of the pendulum and gets embedded in the bob. After collision,if the pendulum moves through a maximum angle of $120^{\circ}$,then the value of $\theta$ is $(g = 10 \text{ ms}^{-2})$.

The potential energy in joules of a particle of mass $1\, kg$ moving in a plane is given by $U = 3x + 4y$,where the position coordinates $x$ and $y$ are measured in metres. If the particle is initially at rest at $(6, 4)$,then:

Which of the following statements is incorrect?

Answer carefully,with reasons:
$(a)$ In an elastic collision of two billiard balls,is the total kinetic energy conserved during the short time of collision of the balls (i.e.,when they are in contact)?
$(b)$ Is the total linear momentum conserved during the short time of an elastic collision of two balls?
$(c)$ What are the answers to $(a)$ and $(b)$ for an inelastic collision?
$(d)$ If the potential energy of two billiard balls depends only on the separation distance between their centres,is the collision elastic or inelastic?
(Note: We are talking here of potential energy corresponding to the force during collision,not gravitational potential energy.)

$A$ 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 . . . . .