Consider a one diamensional elastic collision between a given incoming body A and B, initially at rest. How would you choose the mass of B in comparision to the mass of A in order that B should recoil with
(a) greatest speed
(b) greatest momentum and
(c ) greatest kinetic energy ?

Explanation

As in a collision, momentum is always conserved
`m_(A)u=m_(A)v_(A)+m_(B)v_(B)`
`u=v_(A)+kv_(B)("with k"=m_(B)//m_(A))`
`v_(A)=u-kv_(B) " "` ….(1)
`(1)/(2)m_(A)u^(2)=(1)/(2)m_(A)v_(A)^(2)+(1)/(2)m_(B)v_(B)^(2)`
`u^(2)=v_(A)^(2)+kv_(B)^(2) " "` .....(2)
Substituting the value of `v_(A)` from Eqn. (1) in (2)
`u^(2)=(u-kv_(B))^(2)+kv_(B)^(2)`
Which on solving given `v_(B)=(2u)/(1+k) " "` .....(3)
(a) So for `v_(B)` to be maximum, k must be minimum,
i.e., `k =(m_(B))/(m_(A))to 0` or `m_(B)lt lt m_(A)`
(b) Now `P_(B)=m_(B)v_(B)`
In perfect elastic collisions, when masses are equal, the total momentum of A is completely transfered to B. i.e., `m_(A)=m_(B)`.