A) \[{{\left[ \left( A-1 \right)\left( A+1 \right) \right]}^{2}}\]
B) \[{{\left[ \left( A+1 \right)\left( A-1 \right) \right]}^{2}}\]
C) \[{{\left[ \left( A-1 \right)/A \right]}^{2}}\]
D) \[{{\left[ \left( A+1 \right)/A \right]}^{2}}\]
Correct Answer: A
Solution :
[a] \[v{{'}_{1}}=\frac{\left( {{m}_{1}}-{{m}_{2}} \right){{v}_{1}}+2{{m}_{2}}{{v}_{2}}}{{{m}_{1}}+{{m}_{2}}}\] As \[{{v}_{2}}\] is zero, \[{{m}_{2}}>{{m}_{1}},v{{'}_{1}}\] is in the opposite direction. \[{{m}_{1}}=1,{{m}_{2}}=A.\] \[\therefore \,\left| v{{'}_{1}} \right|=\frac{\left( A-1 \right)}{\left( A+1 \right)}{{v}_{1}}\] The fraction of total energy retained is \[\frac{1/2mv'_{1}^{2}}{1/2v_{1}^{2}}=\frac{\left( A-1 \right)}{\left( A+1 \right)}\]
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