$\begin{array}{1 1} \large\frac{Mv^2(\gamma-1)}{2R} \\ \large\frac{Mv^2(\gamma+1)}{2R} \\ \large\frac{2R}{Mv^2(\gamma-1)} \\ \large\frac{2R}{Mv^2(\gamma+1)}\end{array}$

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- Since internal energy depends only on the temperature, as the temperature changes, the change in internal energy is $\Delta U$= $\large\frac{n\;f\;R\;\Delta T}{2}$

Since internal energy depends only on the temperature, as the temperature changes, the change in internal energy is = $\large\frac{n\;f\;R\;\Delta T}{2}$

Vessel contains a gas of Mass M, change in kinetic energy = $\large\frac{nMv^2}{2}$

$\large\frac{n\;f\;R\;\Delta T}{2} =$ $\large\frac{nMv^2}{2}$

Since, $f = \large\frac{2}{\gamma-1}$ where $\gamma = C_p/C_v$, substituting,we get, $\Delta T = \large\frac{Mv^2(\gamma-1)}{2R}$

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