$A$ particle of mass $m$ collides with a heavy mass $M$ (initially at rest) elastically. After the collision, the particle returns with $4/9$ of its initial kinetic energy. The mass of the heavy object $M$ is ............... $m$.

Discuss elastic collision in two dimensions.

Consider the following statements $A$ and $B$ and identify the correct answer:
$A$. In an elastic collision,if a body suffers a head-on collision with another of the same mass at rest,then the first body comes to rest while the other starts moving with the velocity of the first one.
$B$. Two bodies of equal mass suffering a head-on elastic collision merely exchange their velocities.

Two billiard balls of mass $0.05\,kg$ each,moving in opposite directions with $10\,m/s$,collide and rebound with the same speed. If the time duration of contact is $t = 0.005\,s$,then the force exerted on the balls due to each other is $\dots N$.

$A$ 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: