$A$ body of mass $m$ moving with velocity $v$ undergoes a head-on elastic collision with another body of mass $2m$ which is initially at rest. The ratio of the kinetic energy $(K.E.)$ of the colliding body before and after the collision is: (in $: 1$)

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 a head-on elastic collision with another body $Q$ having mass $m$ initially at rest. If $m << M$,body $Q$ will have a maximum speed equal to $2u$ after the collision.
Reason $R$: During an 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:

It is found that if a neutron suffers an elastic collinear collision with deuterium at rest,the fractional loss of its energy is $P_d$; while for its similar collision with a carbon nucleus at rest,the fractional loss of energy is $P_c$. The values of $P_d$ and $P_c$ are respectively:

$A$ body falling on the ground from a height of $10 \, m$,rebounds to a height of $2.5 \, m$. The ratio of the velocities of the body just before and after the collision will be:

$A$ mass $M$ moving with a certain speed $V$ collides elastically with another stationary mass $m$. After the collision,the masses $M$ and $m$ move with speeds $V^{\prime}$ and $v$,respectively. All motion is in one dimension. Then,

$A$ neutron moving with velocity $u$ collides elastically with a stationary atom of mass number $A$. If the collision is head-on and the initial kinetic energy of the neutron is $E$, then the final kinetic energy of the neutron after the collision is: