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

Consider a one dimensional elastic collision between a given incoming

1.	Consider a one dimensional elastic collision between a given incoming body A and body B, initially at rest. How would you choose the mass of B in comparison to the mass of A in order that B should recoil with (a) greatest speed (b) greatest momentum ?
Answer» Solution :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)`.