In $a$ one-dimensional collision, $a$ particle of mass $2m$ collides with $a$ particle of mass $m$ at rest. If the particles stick together after the collision, what fraction of the initial kinetic energy is lost in the collision?

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?

Three objects $A, B$ and $C$ are kept in a straight line on a frictionless horizontal surface. The masses of ${A}, {B}$ and ${C}$ are ${m}, 2\, {m}$ and $2\, {m}$ respectively. $A$ moves towards ${B}$ with a speed of $9$ ${m} / {s}$ and makes an elastic collision with it. Thereafter $B$ makes a completely inelastic collision with $C.$ All motions occur along same straight line. The final speed of $C$ is $....\,{m} / {s}$

Two particles having position vectors $\overrightarrow {{r_1}} = (3\hat i + 5\hat j)$ metres and $\overrightarrow {{r_2}} = ( - 5\hat i - 3\hat j)$ metres are moving with velocities ${\overrightarrow v _1} = (4\hat i + 3\hat j)\,m/s$ and ${\overrightarrow v _2} = (\alpha \,\hat i + 7\hat j)$ $m/s.$ If they collide after $2$  seconds, the value of $'\alpha '$ is

$Assertion$ : If collision occurs between two elastic bodies their kinetic energy decreases during the time of collision.

$Reason$ : During collision intermolecular space decreases and hence elastic potential energy increases.

In the figure shown, a small ball hits obliquely a smooth and horizontal surface with speed $u$ whose $x$ and $y$ components are indicated. If the coefficient of restitution is $\frac{1}{2}$, then its $x$ and $y$ components $v_x$ and $v_y$ just after collision are respectively