# A ball impinges directly on a similar ball at rest. The first ball is brought to rest by the impact. If half of the kinetic energy is lost by impact, the value of coefficient of restitution is

A) \[\dfrac{1}{{2\sqrt 2 }}\]

B) \[\dfrac{1}{{\sqrt 3 }}\]

C) \[\dfrac{1}{{\sqrt 2 }}\]

D) \[\dfrac{{\sqrt 3 }}{2}\]

Last updated date: 20th Mar 2023

•

Total views: 262.8k

•

Views today: 5.41k

Answer

Verified

262.8k+ views

**Hint:**Here we will use a kinetic energy formula. Also, net change in the kinetic energies will also be calculated.

**Complete answer**:Kinetic Energy: When an object is given some motion then it possesses kinetic energy. Its standard formula is given as, \[{\text{k}} = \dfrac{1}{2}{\text{m}}{{\text{u}}^2}\]where, m is the mass of that object and u is the velocity at which it is moving.

Let the first ball is moving with initial velocity \[{{\text{u}}_1}\], final velocity be \[{{\text{v}}_1}\]. Similarly, the initial velocity of the second ball be \[{{\text{u}}_2}\]and final velocity be \[{{\text{v}}_2}\]. Let the mass of the ball is m.

Now, we will compute the initial kinetic energy and the final kinetic energy of the balls. The formula for total initial kinetic energy is given as,

\[{{\text{K}}_{\text{i}}} = \dfrac{1}{2}{\text{m}}{{\text{u}}_1}^2 + \dfrac{1}{2}{\text{m}}{{\text{u}}_2}^2\]

Total final kinetic energy is given as,

\[{{\text{K}}_{\text{f}}} = \dfrac{1}{2}{\text{m}}{{\text{v}}_1}^2 + \dfrac{1}{2}{\text{m}}{{\text{v}}_2}^2\]

Velocities \[{{\text{u}}_2}\]and\[{{\text{v}}_1}\]are zero. Net change or loss of the kinetic energy is given as,

\[\Delta = \]Total initial kinetic energy –Total final kinetic energy

\[

\Delta = \dfrac{1}{2}{\text{m}}{{\text{u}}_1}^2 + \dfrac{1}{2}{\text{m}}{{\text{u}}_2}^2 - \left( {\dfrac{1}{2}{\text{m}}{{\text{v}}_1}^2 + \dfrac{1}{2}{\text{m}}{{\text{v}}_2}^2} \right) \\

= \dfrac{1}{2}{\text{m}}{{\text{u}}_1}^2 + 0 - 0 - \dfrac{1}{2}{\text{m}}{{\text{v}}_2}^2 \\

\Delta = \dfrac{1}{2}{\text{m}}{{\text{u}}_1}^2 - \dfrac{1}{2}{\text{m}}{{\text{v}}_2}^2 \\

\]

Considering the given question we observe that half of the kinetic energy is lost by impact. Therefore,

Following expression is obtained,

\[ \dfrac{1}{2}\left( {\dfrac{1}{2}{\text{m}}{{\text{v}}_1}^2} \right) = \dfrac{1}{2}{\text{m}}{{\text{u}}_1}^2 - \dfrac{1}{2}{\text{m}}{{\text{v}}_2}^2 \\

{\text{m}}{{\text{v}}_1}^2 = 2{\text{m}}{{\text{u}}_1}^2 - 2{\text{m}}{{\text{v}}_2}^2 \\

{{\text{u}}_1}^2 = 2{{\text{v}}_2}^2 \\

\dfrac{{{{\text{u}}_1}}}{{\sqrt 2 }} = {{\text{v}}_2} \\

\]

Formula of coefficient if restitution is given as,

\[{\text{e}} = \left| {\dfrac{{{{\text{v}}_2} - {{\text{v}}_1}}}{{{{\text{u}}_1} - {{\text{u}}_2}}}} \right|\]Since, the velocities \[{{\text{u}}_2}\] and \[{{\text{v}}_1}\] are zero therefore,

\[\

{\text{e}} = \dfrac{{{{\text{v}}_2}}}{{{{\text{u}}_1}}} \\

{\text{e}} = \dfrac{1}{{\sqrt 2 }} \\

\]

Therefore, the value of the coefficient of restitution is \[\dfrac{1}{{\sqrt 2 }}\].

**Hence, option c is correct.**

**NOTE:**In such types of problems, we must try to apply conservation law of momentum or energy as per the question’s requirement. Also, it is important to learn all the standard formulas of energies like kinetic energy and potential energy.

Recently Updated Pages

Which element possesses the biggest atomic radii A class 11 chemistry JEE_Main

The highly efficient method of obtaining beryllium class 11 chemistry JEE_Main

Which of the following sulphates has the highest solubility class 11 chemistry JEE_Main

Amongst the metal Be Mg Ca and Sr of group 2 of the class 11 chemistry JEE_Main

Which of the following metals is present in the greencolored class 11 chemistry JEE_Main

To prevent magnesium from oxidation in the electrolytic class 11 chemistry JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Change the following sentences into negative and interrogative class 10 english CBSE

A Short Paragraph on our Country India

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

What is the difference between anaerobic aerobic respiration class 10 biology CBSE