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(D) The kinetic energy of the gas is converted into internal energy when the vessel is brought to rest.Let $n$ be the number of moles of the gas.The t

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$\frac{(\gamma - 1)}{2R}Mv^2$

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(D) The kinetic energy of the gas is converted into internal energy when the vessel is brought to rest.Let $n$ be the number of moles of the gas.The total kinetic energy of the gas is $K.E. = \frac{1}{2} (nM) v^2$,where $nM$ is the total mass of the gas.The change in internal energy of an ideal gas is given by $\Delta U = \frac{nR \Delta T}{\gamma - 1}$.Since the vessel is thermally insulated,the kinetic energy is entirely converted into internal energy:$\frac{1}{2} (nM) v^2 = \frac{nR \Delta T}{\gamma - 1}$Canceling $n$ from both sides:$\frac{1}{2} M v^2 = \frac{R \Delta T}{\gamma - 1}$Solving for $\Delta T$:$\Delta T = \frac{M v^2 (\gamma - 1)}{2R}$

$A$ thermally insulated vessel contains an ideal gas of molecular mass $M$ and ratio of specific heats $\gamma$. It is moving with speed $v$ and is suddenly brought to rest. Assuming no heat is lost to the surroundings,its temperature increases by

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