In the previous problem , find the value of `eta` if the the particle of mass `eta m` should recoil with (a) the greater `K.E.` . (b) the greatest momentum and ( c) the greatest speed.

Explanation

(a) `K.E.` of mas `eta m`
`K_(2) = (1)/(2) eta m v_(2)^(2) = (1)/(2) eta m ((2 u)/(1 + eta))^(2) = ((1)/(2) m u^(2))(4 eta)/(( eta + 1)^(2))`
For `K_(2)` to be maximum
`(dK_(2))/( d eta) = 0 rArr ((eta + 1)^(2) xx 4 - 4 eta xx 2(eta + 1))/((eta + 1)^(4)) = 0`
`eta = 1`
OR
`K_(2) = (1)/(2) mu^(2) ( 4 eta)/((eta + 1)^(2)) = (1)/(2) m u^(2) ( 4 eta)/((1 - eta)^(2) + 4 eta)`
For `K_(2)` to be maximum , denominator should be minimum , for this
`1 - eta = 0 rArr eta = 1`
(b) Momentum of mass `eta m`
`p_(2) = eta mv_(2) = eta m (2 u)/((1 + eta))`
`= (2 m u)/(1 + (1)/(eta))`
For `p_(2)` to be maximum
`( 1 + (1)/(eta))` should be minimum , for this
`eta rarr oo` , i.e. mass of the second particle gtgt mass of the first particle
( c) Velocity of mass `eta m`
`v_(2) = ( 2u)/( 1 + eta)`
For `v_(2)` to be maximum `( 1 + eta)` should be minimum , for this , `eta rarr 0` , i.e. mass of the second particle ltlt mass of the first particle