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Two particles of mass m & 2m have veloci...

Two particles of mass m & 2m have velocities `u hat( j) ` & `u hat( i ) ` respectively . They collide completely inelastically. Find the loss in kinetic energy of the system.

A

`m u^(2)`

B

`(2)/(3) m u^(2)`

C

`( m u^(2))/( 4)`

D

`( m u^(2))/( 4)`

Text Solution

Verified by Experts

For the system `overset(1) (P) _(i)= 2 m u hat(i) + m u hat(j)`
As all the impulses are internal , `overset(1) (P)_(i) = overset(1)(P) _(f)`
`overset(1) (P) _(f) = 3 m overset(r ) (u) _(f)` . Where `overset(r ) ( u ) _(f)` is the velocity of the combined mass after inelastic collisoion
Thus `3 m overset(r ) ( u ) _(f) = 2 m u hat(i) + m u hat( j), ` `overset( r ) ( u ) _(f)= (2u )/( 3) hat(i) +(1)/( 3) hat(j) rArr| overset(r )( u ) _(f) | = sqrt((5)/(9)) u `
Thus final energy of the system `= (1)/(2) xx 3m xx | overset( r )( u)_(f) |^(2) = ( 1)/( 2) xx(5)/(9)u^(2) = (5m u^(2))/( 6)`
Initial energy of the system `= (m_(1)v_(1)^(2))/( 2) +( m_(2) v_(2)^(2))/( 2) = ( 1)/( 2) m u^(2) +( 1)/( 2) xx 2m xx u^(2) = ( 3)/( 2) m u ^(2)`
Loss in K.E. `= [` Initial K.E. ) - ( Final K.E. )]
Loss `= (3)/( 2) m u^(2) - ( 5)/(6) m u^(2) = (2)/( 3) m u^(2)`
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