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, monentum 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)`
Now as collision is elastic `(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)`
Subsituting the value of `v_(A)` from Eqn. (1) in (2)
`u^(2)=(u-kv_(B))^(2)+kv_(B)^(2)`
Which on solving gives `v_(B) = (2u)/(i+k) " " .......(3)`
(a) So for `v_(B)` to be maximum, k must be minimum,
`i.e., k=(m_(B))/(m_(A))rarr0orm_(B)ltltm_(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)`.