In a collinear collision, a particle with an initial speed $v_0$ strikes a stationary particle of the same mass. If the final total kinetic energy is $50\%$ greater than the original kinetic energy, the magnitude of the relative velocity between the two particles after collision is

Statement-I: If there is no external torque on a body about its centre of mass, then the velocity of the centre of mass remains constant.

Statement-2: The linear momentum of an isolated system remains constant.

Two thin circular discs of mass $\mathrm{m}$ and $4\mathrm{m},$having radii of $a$ and $2a,$respectively, are rigidly fixed by a massless, rigid rod of length $l=\sqrt{24}a$ through their centers. This assembly is laid on a firm and flat surface, and set rolling without slipping on the surface so that the angular speed about the axis of the rod is $\omega$. The angular momentum of the entire assembly about the point$\text{'}O\text{'}$is $\overrightarrow{\mathrm{L}}$ (see the figure). Which of the following statement (s) is (are) true ?

A particle of mass $m_1$ experienced a perfectly elastic collision with a stationary particle of mass $m_2$. What fraction of the kinetic energy does the striking particle lose, if 

$\left(a\right)$ it recoils at right angles to its original motion direction, 

$\left(b\right)$ the collision is a head-on one?

Particle $1$ experiences a perfectly elastic collision with a stationary particle $2$. Determine their mass ratio, if

$\left(a\right)$ after a head-on collision the particles fly apart in the opposite directions with equal velocities, 

$\left(b\right)$ the particles fly apart symmetrically relative to the initial motion direction of particle 1 with the angle of divergence $\text{θ}=60°$.