Two billiard balls undergo a head-on collision. Ball $1$ is twice as heavy as ball $2$. Initially, ball $1$ moves with a speed $v$ towards ball $2$ which is at rest. Immediately after the collision, ball $1$ travels at $a$ speed of $v/3$ in the same direction. What type of collision has occured?

Given below are two statements: one is labelled as Assertion $A$ and the other is labelled as Reason $R$.

Assertion $A$ : Body $'P'$ having mass $M$ moving with speed $'u'$ has head-on collision elastically with another body $'Q'$ having mass $'m'$ initially at rest. If $m< < M,$ body $'Q'$ will have a maximum speed equal to $'2u'$ after collision.

Reason $R$ : During elastic collision, the momentum and kinetic energy are both conserved.

In the light of the above statements, choose the most appropriate answer from the options given below:

A particle of mass $m$ moving with a speed $v$ hits elastically another stationary particle of mass $2\ m$ in a fixed smooth horizontal circular tube of radius $R$. Find the time when the next collision will take place?

Two particles $A$ and $B,$ move with constant  velocities $\vec v_1$ and $\vec v_2$. At the initial moment their position vectors are $\vec r_1$ and $\vec r_2$ respectively. The condition for particles $A$ and $B$ for their collision is

Consider elastic collision of a particle of mass $m $ moving with a velocity $u$ with another particle of the same mass at rest. After the collision the projectile and the struck particle move in directions making angles ${\theta _1}$and ${\theta _2}$respectively with the initial direction of motion. The sum of the angles. ${\theta _1} + {\theta _2},$ is......$^o$

The bob $A$ of a pendulum released from $30^o$ to the vertical hits another bob $B$ of the same mass at rest on a table as shown in Figure. How high does the bob A rise after the collision ? Neglect the size of the bobs and assume the collision to be elastic