A `He^(+)` ion is at rest and is in ground state. A necutron with intial kinetic energy`K` collides heads on with the `He^(+)` ion. Find minimum value of `K` so that there can be an inelastic collision between these two particals.

Explanation

Here the loss during the collision can only be used to excite the atoms or electrons. So according to quantum mechanics
loss `={0,40.8eV,48.3eV,…..54.4eV}`…..(1)
`E_(n)= -13.6(Z^(2))/(n^(2))eV`

Now according to newtonion mechanics Minimum loss `=0`
maximum loss will be for perfectly inelastic collision, let `v_(0)` be the initial speed of neutron and `v_(f)` be the final common speed.
so by momentum consrvation `mv_(0)=mv_(f)+4mv_(f) v_(f)=(v_(0))/(5)`
where `m=` mass of Neutron
`:.` mass of `He^(+)` ion `=4m`
so final kinetic energy of system
`K.E=(1)/(2)m v_(f)^(2)+(1)/(2)4mv_(f)^(2)=(1)/(2).(5cm).(v_(0)^(2))/(25)=(1)/(5).((1)/(2)mv_(0)^(2))=(K)/(5)`
maximum loss `=K-(K)/(5)=(4K)/(5)`
so loss will be `[0,(4K)/(5)]`.......(2)
for inelastic collsion there should be at least one common value other than zero in set (1) and (2)
`:. (4K)/(5) gt 40.8eV`
`K gt 51 eV`
minimum value of `K=51eV`.