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

Answer

Verified

318.9k+ 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.

Last updated date: 25th Sep 2023

•

Total views: 318.9k

•

Views today: 8.18k

Recently Updated Pages

10 Examples of Evaporation in Daily Life with Explanations

10 Examples of Friction in Our Daily Life

What do you mean by public facilities

Difference between hardware and software

Disadvantages of Advertising

10 Advantages and Disadvantages of Plastic

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

What is the IUPAC name of CH3CH CH COOH A 2Butenoic class 11 chemistry CBSE

Drive an expression for the electric field due to an class 12 physics CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

The dimensions of potential gradient are A MLT 3A 1 class 11 physics CBSE

Define electric potential and write down its dimen class 9 physics CBSE

Why is the electric field perpendicular to the equipotential class 12 physics CBSE