Question

Hello friends !!

A ball of mass m moving at a speed v makes a head - on collision with an identical ball at rest . the kinetic energy of the balls after the collision is three fourths of the original. find the cofficient of restitution .
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Answer 1

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ANSWER::


Mass of first ball = m

Speed of first ball = v

Mass of second ball = m


Let final velocities of first and second ball are v₁ and v₂ respectively.


Using law of conservation of momentum,


m(v₁ + v₂) = mv


v₁ + v₂ = v Equation 1


And , v₁ - v₂ = ev Equation 2


Given:-


Final Kinetic Energy = 3/4 Initial Kinetic Energy


(1/2)mv₁² + (1/2)mv₂² = (3/4) x 1/2 mv²


v₁² + v₂² = (3/4) v²


[(v₁ + v₂)² + (v₁ - v₂)² ] / 2 = 3v² / 4


(1 + e²)v² / 2 = 3v² / 4


1 + e² = 3/2


e² = 1/2


e = 1 / √2


Hope it helps!

Answer 2

$\star \bold \blue{hello} \star$

$\bold \green{solution}$:-

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$\bold \orange{given}$:-

mass of 1st ball is m speed = v

mass of 2nd ball is m let final velocity of 1st ball and 2nd ball is v1 and v2 respectively

.*using colm *__________m ( v1 + v2) = mv v1 + v2 = v ____(1)

v1 - v2 = ev _______(2)

Given that final K.E = 3/4

initian K.E 1/2mv1² + 1/2mv2² = 3/4 * 1/2 mv² v1² + v2² = 3/4v²

or,( v1 + v2 )² + (v1 - v2 )² /2 = 3/4 v²

=> ( 1 + e²)v² /2 = 3/4 v²

=> 1 + e² = 3/2

=> e² = 3/2 - 1

=> e = 1/√2 Answer ✔

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