A) \[{{\left( \frac{A-1}{A+1} \right)}^{2}}\]
B) \[{{\left( \frac{A+1}{A-1} \right)}^{2}}\]
C) \[{{\left( \frac{A-1}{A} \right)}^{2}}\]
D) \[{{\left( \frac{A+1}{A} \right)}^{2}}\]
Correct Answer: A
Solution :
The general formula of final velocity of the first body after the collision is, \[{{V}_{1}}=\frac{{{m}_{1}}-{{m}_{2}}}{{{m}_{1}}+{{m}_{2}}}{{u}_{1}}+\frac{2{{m}_{2}}{{u}_{2}}}{{{m}_{1}}+{{m}_{2}}}\] For \[{{u}_{2}}=0\], \[{{V}_{1}}=\frac{{{m}_{1}}-{{m}_{2}}}{{{m}_{1}}+{{m}_{2}}}{{u}_{1}}\] Thus, \[\frac{{{V}_{1}}}{{{u}_{1}}}=\frac{{{m}_{1}}-{{m}_{2}}}{{{m}_{1}}+{{m}_{2}}}\] Now, fraction of the kinetic energy retained \[=\frac{Final\,\,kinetic\,\,in\,\,the\,\,first\,\,object}{Initial\,\,kinetic\,\,energy}\] \[=\frac{\frac{1}{2}{{m}_{1}}v_{1}^{2}}{\frac{1}{2}{{m}_{1}}u_{1}^{2}}=\frac{{{v}_{{{1}^{2}}}}}{u_{1}^{2}}\] \[={{\left( \frac{{{m}_{1}}-{{m}_{2}}}{{{m}_{1}}+{{m}_{2}}} \right)}^{2}}\] Given that:\[{{m}_{1}}={{m}_{2}}=A\]\[,\]\[{{u}_{1}}=v\]\[,\] \[{{u}_{2}}=0\] Substituting, these in the above equation, we have Fraction of kinetic energy retained\[={{\left( \frac{1-A}{1+A} \right)}^{2}}\] \[={{\left( \frac{A-1}{A+1} \right)}^{2}}\]
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