Two small particles of equal masses start moving in opposite directions from a point $A$ in a horizontal circular orbit. Their tangential velocities are $v$ and $2v$,respectively,as shown in the figure. Between collisions,the particles move with constant speeds. After making how many elastic collisions,other than that at $A$,will these two particles again reach the point $A$?

$A$ body of mass $5 \ kg$ makes an elastic collision with another body at rest and continues to move in the original direction after collision with a velocity equal to $\frac{1}{10}$ th of its original velocity. Then the mass of the second body is (in $kg$)

$A$ particle with kinetic energy $K = 3 \ J$ makes an elastic head-on collision with a stationary particle of twice its mass. Which of the following statements is correct?

$A$ sphere of mass $0.1\,kg$ is attached to a cord of $1\,m$ length. Starting from the height of its point of suspension,this sphere hits a block of the same mass at rest on a frictionless table. If the impact is elastic,then the kinetic energy of the block after the collision is ............. $J$.

$A$ particle of mass $m$ moving with a velocity $u$ makes an elastic one-dimensional collision with a stationary particle of mass $m$. They are in contact for a total time $T$. The contact force increases linearly from $0$ to $F_0$ in time $\frac{T}{4}$, remains constant for a further time $\frac{T}{2}$, and decreases linearly from $F_0$ to $0$ in the final time $\frac{T}{4}$, as shown in the graph. The magnitude of $F_0$ is:

Given below are two statements. One is labelled as Assertion $(A)$ and the other is labelled as Reason $(R)$. Three identical spheres of same mass undergo one-dimensional motion as shown in the figure with initial velocities $v_{A} = 5 \ m/s$,$v_{B} = 2 \ m/s$,$v_{C} = 4 \ m/s$. If we wait sufficiently long for elastic collisions to happen,then $v_{A} = 4 \ m/s$,$v_{B} = 2 \ m/s$,$v_{C} = 5 \ m/s$ will be the final velocities.
Reason $(R)$: In an elastic collision between identical masses,two objects exchange their velocities. In the light of the above statements,choose the correct answer from the options given below: