A) \[{{\left( \frac{1}{2} \right)}^{\frac{\gamma +1}{2}}}\]
B) \[{{\left( \frac{1}{2} \right)}^{\gamma }}\]
C) \[2\]
D) \[\frac{1}{2}\]
Correct Answer: A , B , C , D
Solution :
(BONUS) \[\tau \propto \frac{1}{n<v>},\,<v>\,\propto \sqrt{T}\] \[\Rightarrow \,\,\,\tau \propto \frac{1}{n\sqrt{T}}\Rightarrow \,\frac{{{\tau }_{2}}}{{{\tau }_{1}}}=\frac{{{n}_{1}}}{{{n}_{2}}}\sqrt{\frac{{{T}_{1}}}{{{T}_{2}}}}\] \[=2\sqrt{\frac{{{T}_{1}}}{{{T}_{2}}}}\] \[{{T}_{1}}{{V}_{1}}^{\gamma -1}={{T}_{2}}{{(2{{V}_{1}})}^{\gamma -1}}\Rightarrow \,\frac{{{T}_{1}}}{{{T}_{2}}}={{2}^{\gamma -1}}\] \[\Rightarrow \,\,\frac{{{\tau }_{2}}}{{{\tau }_{1}}}\,=2\times {{2}^{\frac{(\gamma -1)}{2}}}=2{{\,}^{\left( \frac{\gamma +1}{2} \right)}}\] NOTE: Answer does not match with given options.Solution :
(BONUS) \[\tau \propto \frac{1}{n<v>},\,<v>\,\propto \sqrt{T}\] \[\Rightarrow \,\,\,\tau \propto \frac{1}{n\sqrt{T}}\Rightarrow \,\frac{{{\tau }_{2}}}{{{\tau }_{1}}}=\frac{{{n}_{1}}}{{{n}_{2}}}\sqrt{\frac{{{T}_{1}}}{{{T}_{2}}}}\] \[=2\sqrt{\frac{{{T}_{1}}}{{{T}_{2}}}}\] \[{{T}_{1}}{{V}_{1}}^{\gamma -1}={{T}_{2}}{{(2{{V}_{1}})}^{\gamma -1}}\Rightarrow \,\frac{{{T}_{1}}}{{{T}_{2}}}={{2}^{\gamma -1}}\] \[\Rightarrow \,\,\frac{{{\tau }_{2}}}{{{\tau }_{1}}}\,=2\times {{2}^{\frac{(\gamma -1)}{2}}}=2{{\,}^{\left( \frac{\gamma +1}{2} \right)}}\] NOTE: Answer does not match with given options.Solution :
(BONUS) \[\tau \propto \frac{1}{n<v>},\,<v>\,\propto \sqrt{T}\] \[\Rightarrow \,\,\,\tau \propto \frac{1}{n\sqrt{T}}\Rightarrow \,\frac{{{\tau }_{2}}}{{{\tau }_{1}}}=\frac{{{n}_{1}}}{{{n}_{2}}}\sqrt{\frac{{{T}_{1}}}{{{T}_{2}}}}\] \[=2\sqrt{\frac{{{T}_{1}}}{{{T}_{2}}}}\] \[{{T}_{1}}{{V}_{1}}^{\gamma -1}={{T}_{2}}{{(2{{V}_{1}})}^{\gamma -1}}\Rightarrow \,\frac{{{T}_{1}}}{{{T}_{2}}}={{2}^{\gamma -1}}\] \[\Rightarrow \,\,\frac{{{\tau }_{2}}}{{{\tau }_{1}}}\,=2\times {{2}^{\frac{(\gamma -1)}{2}}}=2{{\,}^{\left( \frac{\gamma +1}{2} \right)}}\] NOTE: Answer does not match with given options.Solution :
(BONUS) \[\tau \propto \frac{1}{n<v>},\,<v>\,\propto \sqrt{T}\] \[\Rightarrow \,\,\,\tau \propto \frac{1}{n\sqrt{T}}\Rightarrow \,\frac{{{\tau }_{2}}}{{{\tau }_{1}}}=\frac{{{n}_{1}}}{{{n}_{2}}}\sqrt{\frac{{{T}_{1}}}{{{T}_{2}}}}\] \[=2\sqrt{\frac{{{T}_{1}}}{{{T}_{2}}}}\] \[{{T}_{1}}{{V}_{1}}^{\gamma -1}={{T}_{2}}{{(2{{V}_{1}})}^{\gamma -1}}\Rightarrow \,\frac{{{T}_{1}}}{{{T}_{2}}}={{2}^{\gamma -1}}\] \[\Rightarrow \,\,\frac{{{\tau }_{2}}}{{{\tau }_{1}}}\,=2\times {{2}^{\frac{(\gamma -1)}{2}}}=2{{\,}^{\left( \frac{\gamma +1}{2} \right)}}\] NOTE: Answer does not match with given options.
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