A particle of mass `m _(1)` makes a head on elastic collision with a stationary particle of mass `m _(2).` What fraction of kinetic energy is
(a) retained by `m _(1)`
(b) transferred onto `m _(2)` ?

Explanation

(a) ` v _(1) = (( m _(1) - m _(2)) u + 2m _(2) (o))/( m _(1) + m _(2)) implies (( v _(1))/(u ))^(2) = ((m_(1) - m _(2))/(m _(1) + m _(2))) ^(2)`
`implies ((1)/(2) m _(1) v _(1) ^(2))/( (1)/(2) m _(1) u ^(2)) = ((m _(1) - m _(2)) /( m _(1) + m _(2))) ^(2) (KE of 1 ^(st ) " particle finally")/(KE of 1 ^(st) "particle initially") = (( m _(1) - m _(2))/( m _(1) + m _(2))) ^(2)`
`therefore` fraction of KE retained by `1 ^(st)` body `f _(1) = ((m _(1) - m _(2))/( m _(1) + m _(2)) ) ^(2)`
(b) `v _(2) = (2 m _(1) u + ( m _(2) - m _(1)) 0o)/(m _(1) + m_(2)) implies(v _(2))/( u ) = (2m _(1))/( m _(1) + m _(2))`
`f _(2) = ((1)/(2) m _(2) v _(2) ^(2))/( (1)/(2) m _(1) u ^(2))= (m_(2))/( m _(1)) [ ( 2m _(1))/( m _(1) + m _(2)) ] ^(2)`
`therefore f _(2) = ( 4m _(1) m _(2))/( ( m _(1) + m _(2)) ^(2))` (fraction of KE given to `2 ^(nd)` body)