A) \[\frac{h}{{{m}_{A}}V}\left[ \left| \frac{{{m}_{A}}+{{m}_{B}}}{{{m}_{A}}-{{m}_{B}}} \right|-1 \right]\]
B) \[\frac{h}{{{m}_{B}}V}\left[ \left| \frac{{{m}_{A}}+{{m}_{B}}}{{{m}_{A}}-{{m}_{B}}} \right|-1 \right]\]
C) \[\frac{h}{{{m}_{A}}V}\left[ \left| \frac{{{m}_{A}}+{{m}_{B}}}{{{m}_{A}}+{{m}_{B}}} \right|-1 \right]\]
D) \[\frac{h}{{{m}_{B}}V}\left[ \left| \frac{{{m}_{A}}-{{m}_{B}}}{{{m}_{A}}+{{m}_{B}}} \right|-1 \right]\]
Correct Answer: A
Solution :
[a] From the law of conservation of momentum \[{{m}_{A}}V={{m}_{A}}{{V}_{A}}+{{m}_{B}}{{V}_{B}}\] Or \[{{m}_{A}}(V-{{V}_{A}})={{m}_{B}}{{V}_{B}}\] ? (i) \[\frac{1}{2}{{m}_{A}}{{V}^{2}}=\frac{1}{2}{{m}_{A}}{{V}_{A}}^{2}+\frac{1}{2}{{m}_{B}}{{V}_{B}}^{2}\] \[{{m}_{A}}\left( {{V}^{2}}-{{V}_{A}}^{2} \right)={{m}_{B}}{{V}_{B}}^{2}\] Dividing eqn. (ii) by eqn. (i), we obtain Or \[\frac{V}{{{V}_{A}}}=\left( \frac{{{m}_{A}}+{{m}_{B}}}{{{m}_{A}}-{{m}_{B}}} \right)\] ???(iv) Change in wavelength \[\Delta \lambda ={{({{\lambda }_{A}})}_{f}}-({{\lambda }_{A}})=\frac{h}{{{m}_{A}}{{V}_{A}}}-\frac{h}{{{m}_{A}}V}\] Or \[\Delta \lambda =\frac{h}{{{m}_{A}}V}-\left[ \frac{V}{{{V}_{A}}}-1 \right]\] ???(v) From eqn. (iv) and (v). \[\Delta \lambda =\frac{h}{{{m}_{A}}V}\left[ \left| \frac{{{m}_{A}}+{{m}_{B}}}{{{m}_{A}}-{{m}_{B}}} \right|-1 \right]\]
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