Explanation
The law of conservation of linear momentum tells us that the overall momentum before the collision must be equal to the overall momentum after a collision.
Since the spheres have identical masses, we can write
$$mu + m\times 0 = mv_A + mv_B$$
$$u = v_A+v_B$$
From the definition of the coefficient of restitution, we know that
$$e = \dfrac{v_B - v_A}{u}$$
solving above two equations
$$e \times ( v_A+v_B) = v_B - v_A$$
$$v_B(1-e) = v_A (1+e)$$
$$\dfrac{v_A}{v_B} = \dfrac{1-e}{1+e}$$
$$\textbf{Hint}$$: In both the elastic and inelastic collisions, the momentum of the system remains conserved.
$$\textbf{Explanation:}$$
(A) Since in the collision of the bodies, the momentum of the system is always conserved, therefore option (A) is not possible because then the final momentum will be zero, which is not possible.
(B) After the collision of the bodies, the velocity transfers according to the mass of the bodies, the velocity transferred is inversely proportional to the mass of the body, and hence both the body can move after the collision and this option is correct.
(C) If the mass of the bodies will be equal then since the momentum of the system is conserved hence the velocity of the moving body will be transferred to the resting body and it can move with the same velocity as the first body and the first body will come to rest. Hence this option is correct.
(D) This option is not possible because then the momentum of the system will change.
$$\textbf{Thus, options (B) and (C) are correct.}$$
Hint: The linear momentum of a particle is defined as the product of the mass of the particle times the velocity of that particle.
Step 1: Formulas and concepts used:
We know that $${v_1} < v$$
By coefficient of restitution (e) formula,
$$v \times e = {v_2} - {v_1}$$
The system's momentum, rather than the individual particles', is preserved. Individual bodies in the system's momentum may grow or decrease depending on the environment, but the system's momentum will always be preserved as long as no external net force is exerted on it.
Formula is given by,
Initial momentum=final momentum
$${p_i} = {p_f}$$
Step 2: Substituting values:
Substituting values of $${v_2}$$,
$${m_1}v = {m_2}({ve} + {v_1}) - {m_1}{v_1}$$
$${m_1}v = {m_2}{ve} + {m_2}{v_1} - {m_1}{v_1}$$
$${m_1}(v + {v_1}) = {m_2}({ve} + {v_1})$$
$$\dfrac{{{m_1}}}{{{m_2}}} = \dfrac{{({ve} + {v_1})}}{{(v + {v_1})}}$$
As $${ve} \leqslant v$$ $$ \therefore e \leq {1}$$
$$\dfrac{{{m_1}}}{{{m_2}}} \leqslant 1$$
$$\therefore {m_1} < {m_2}$$
Hence, after collision $$ {m_1} < {m_2}$$Correct answer: Option C
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