Question

A block having mass \[m\]collides with another stationary block having mass \[2\,m.\]The lighter block comes to rest after collision. If the initial velocity of the first block is \[v,\]then the value of coefficient of restitution will be?
A. 0.5
B. 0.4
C. 0.6
D. 0.8

Answer 1

Hint: In this question we will proceed with the help of Law of conservation of momentum inelastic collision to find the value of final velocity in terms of initial velocity and then we will find the coefficient of restitution by given formula.

Using Formula: -
Law of conservation of momentum:
According to Law of conservation of momentum the linear momentum of any isolated system (where there is no external force applied on the system) is conserved so
\[{m_1}{u_1} + {m_2}{u_2} = {m_1}{v_1} + {m_2}{v_2}\]
Where \[{m_1},\,{m_2}\]are masses of the two bodies
\[{u_1},{u_2}\]are the initial velocities of first body and second body respectively
\[{v_1},{v_2}\]are the final velocities of first body and second body respectively
Coefficient of restitution: It is defined as the ratio of velocity of separation to the velocity of approach before and after the inelastic collision of two bodies
 \[e = \dfrac{{({v_2} - {v_1})}}{{({u_1} - {u_2})}}\]
Given Data: -
$ {u_1} = v \\
  {u_2} = 0 \\
  {v_1} = 0 \\
  {v_2} = ? \\
  {m_1} = m \\
  {m_2} = 2m \\
  e = ? \\ $

Complete step-by-step answer:
Using the law of conservation of momentum, we have
${m_1}{u_1} + {m_2}{u_2} = {m_1}{v_1} + {m_2}{v_2}$
 $ m \times v + 2m \times 0 = m \times 0 + 2m \times {v_2} \\$
  $mv = 2m{v_2} \\$
  ${v_2} = \dfrac{{mv}}{{2m}} = \dfrac{v}{2} \\$
Now the coefficient of restitution is given by
$e = \dfrac{{({v_2} - {v_1})}}{{({u_1} - {u_2})}}$
$e = \dfrac{{v/2 - 0}}{{v - 0}} = \dfrac{{v/2}}{v} \\$
$e = 0.5 \\$
In this way the required coefficient of restitution of the system of bodies given in the question is equal to $0.5$
Hence the correct option is\[(A)\].

Note: It is important to note that we should be aware of the application of the Law of conservation of momentum and inelastic collision. In inelastic collisions there is a loss of kinetic energy after collision of the bodies.