Two particles of equal mass m have respective initial velocities u ^ i and u( ^ i + ^ j 2 ) . They collide completely inelastically. The energy lost in the process is:

Correct answer: B. (1)/(8)m u ^ 2 Conserving momentum mu ^ i + m((u)/(2) ^ i + (u)/(2) ^ j ) = 2m( u _ 1 ^ i + u _ 2 ^ j ) On solving u _ 1 = (3u)/(4) and u _ 2 = (u)/(4) Change in K.E. [(1)/(2)m u ^ 2 + (1)/(2)m ((u)/(2) 2 ) ^ 2 ] - [(1)/(2)(2m)( 9 u ^ 2 16 + u ^ 2 16 )] = 3m u ^ 2 4 - 5m u ^ 2 8 = m u ^ 2 8

JEE Main 20209 JanuaryMorning Shift

Exam: JEE Main
Year: 2020
Date: 9 January
Shift: Morning Shift
Subject: Physics
Chapter: Center of Mass, Momentum and Collision

JEE Main 2020 · 9 January · Morning Shift · Physics · Center of Mass, Momentum and Collision

Two particles of equal mass m have respective initial velocities u...

Two particles of equal mass

m

have respective initial velocities

u\overset{^}{i}

and u\left(\dfrac{\overset{^}{i} + \overset{^}{j}}{2}\right) . They collide completely inelastically. The energy lost in the process is:

Options

A $\dfrac{1}{3}m{u}^{2}$
B $\dfrac{1}{8}m{u}^{2}$
C $\dfrac{3}{4}m{u}^{2}$
D $\sqrt{\dfrac{2}{3}}m{u}^{2}$

Correct answer

B. $\dfrac{1}{8}m{u}^{2}$

Step-by-step solution

Conserving momentum

mu\overset{^}{i} + m\left(\dfrac{u}{2}\overset{^}{i} + \dfrac{u}{2}\overset{^}{j}\right) = 2m\left({u}_{1}\overset{^}{i} + {u}_{2}\overset{^}{j}\right)

On solving

{u}_{1} = \dfrac{3u}{4}

and

{u}_{2} = \dfrac{u}{4}

Change in K.E.
\left[\dfrac{1}{2}m{u}^{2} + \dfrac{1}{2}m{

\left(\dfrac{u}{2}\sqrt{2}\right)}^{2}\right] - \left[\dfrac{1}{2}\left(2m\right)\left(\dfrac{9{u}^{2}}{16} + \dfrac{{u}^{2}}{16}\right)\right]

= \dfrac{3m{u}^{2}}{4} - \dfrac{5m{u}^{2}}{8} = \dfrac{m{u}^{2}}{8}

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