A thermally insulated vessel containing diatomic gas of molar mass M is moving with velocity v. the temperature of gas is T. if it is suddenly stopped, then

A. Internal energy of gas is equal to `(5nRT)/(2)` when vessel is moving
B. There is no change in temperature if vessel is suddenly stopped
C. There is rise in temperature if vessel is suddenly stopped
D. Temperature increase is due to heat input

Options

A. a. Internal energy of gas is equal to `(5nRT)/(2)` when vessel is moving
B. b. There is no change in temperature if vessel is suddenly stopped
C. c. There is rise in temperature if vessel is suddenly stopped
D. d. Temperature increase is due to heat input

Answer

To solve the problem, we need to analyze the situation of a thermally insulated vessel containing a diatomic gas that is initially moving with a velocity \( v \) and has a temperature \( T \). When the vessel is suddenly stopped, we need to determine the effects on the internal energy and temperature of the gas.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- The vessel is thermally insulated, meaning no heat can enter or leave the system.
- The gas inside the vessel is diatomic with a molar mass \( M \).
- The gas is moving with velocity \( v \) and has a temperature \( T \).

2. **Internal Energy of the Gas**:
- The internal energy \( U \) of a diatomic gas is given by the formula:
\[ U = \frac{5}{2} nRT \]
where \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature.
- While the vessel is moving, the gas has kinetic energy due to its motion. The total energy of the gas includes both its internal energy and the kinetic energy due to its motion.

3. **Kinetic Energy Before Stopping**:
- The kinetic energy \( KE \) of the gas when the vessel is moving is:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the gas. Since \( m = nM \) (where \( M \) is the molar mass), we can express it as:
\[ KE = \frac{1}{2} nM v^2 \]

4. **Effect of Stopping the Vessel**:
- When the vessel is suddenly stopped, the kinetic energy of the gas is converted into internal energy due to the collisions of gas molecules.
- The change in kinetic energy will lead to an increase in the internal energy of the gas.

5. **Temperature Change**:
- Since the vessel is thermally insulated, no heat is exchanged with the surroundings. The increase in internal energy translates to an increase in temperature.
- The relationship between internal energy and temperature for a diatomic gas indicates that the temperature will rise when kinetic energy is converted to internal energy.

6. **Conclusion**:
- Therefore, when the vessel is suddenly stopped, there is a rise in temperature due to the conversion of kinetic energy to internal energy. The internal energy of the gas when it was moving is indeed \( \frac{5}{2} nRT \).
- The correct options based on the analysis are:
- A: Internal energy of gas is equal to \( \frac{5nRT}{2} \) when the vessel is moving (True).
- C: There is a rise in temperature if the vessel is suddenly stopped (True).
- D: Temperature increase is due to heat input (True, in the sense that the kinetic energy conversion acts like heat input).

### Final Answer:
- The correct options are A, C, and D.