A particle A with a mass `m_(A)` is moving with a velocity v and hits a particle B (mass `m_(B)`) at rest (one dimensional motion). Find the change in the de-Broglie wavelength of the particle A. Treat the collision as elastic.

Options

A. `(h)/(m_(A)v)[|(m_(A)+m_(B))/(m_(A)-m_(B))|-1]`
B. `(h)/(m_(B)v)[|(m_(A)+m_(B))/(m_(A)-m_(B))|-1]`
C. `(h)/(m_(A)v)[|(m_(A)-m_(B))/(m_(A)+m_(B))|-1]`
D. `(h)/(m_(B)v)[|(m_(A)-m_(B))/(m_(A)+m_(B))|-1]`

Answer

A

Explanation

From the law of conservation of momentum
`m_(A)v=m_(A)v_(B)`
or `m_(A)(v-v_(A))=m_(B)v_(B)" "...(i)`
`(1)/(2)m_(A)v^(2)=(1)/(2)m_(A)v_(A)^(2)+(1)/(2)m_(B)v_(B)^(2)`
`m_(A)(v^(2)-V_(A)^(2))=m_(B)v_(B)^(2)`
Dividing eqn. (ii) by eqn. (i), we obtain
or `(v)/(v_(A))=((m_(A)+m_(B))/(m_(A)-m_(B)))" "...(iv)`
Change in wavelength
`Deltalambda=(lamda_(A))_("f")-(lamda_(A))=(h)/(m_(A)v_(A))-(h)/(m_(A)v)`
or `Deltalamda=(h)/(m_(A)v)[(v)/(v_(A))-1]" "...(v)`
From eqn. (iv) and (v).
`Deltalamda=(h)/(m_(A)v)[|(m_(A)+m_(B))/(m_(A)-m_(B))|-1]`