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) 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)` ....(2) Substituting the value of `v_(A)` from Eqn. (1) and (2) `u^(2)= (u-kv_(B))^(2)+ kv_(B)^(2)` Which on solving gives `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))rarr 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)`.