A frictionless track $ABCDE$ ends in a circular loop of radius $R$ .A body slides down the track from point $A$ which is at a height $h = 5\, cm$. Maximum value of $R$ for the body to successfully complete the loop is .................. $\mathrm{cm}$

The potential energy of a body of mass $m$ is:
                      $U = ax + by$
Where $x$ and $y$ are position co-ordinates of the particle. The acceleration of the particle is

$A$ particle of mass $m$ is released from $a$ height $H$ on $a$ smooth curved surface which ends into a vertical loop of radius $R$, as shown The minimum value of $H$ required so that the particle makes a complete vertical circle is given by

Figure shows the vertical section of frictionless surface. A block of mass $2\, kg$ is released from the position $A$ ; its $KE$ as it reaches the position $C$ is ............ $\mathrm{J}$

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).

Two identical $5\,\,kg.$  blocks are moving with same speed of $2\,\,m/s$  towards each other along a frictionless horizontal surface. The two blocks collide, stick together and come to rest. Consider to two blocks as a system, the work done on the system by the external forces will be .............. $\mathrm{Joule}$