A cubical box containing a gas with internal energy U is given velocity V, then the new internal energy of the gas

Options

A. less than U
B. more than U
C. U
D. zero

Answer

To determine the new internal energy of a gas contained in a cubical box that is given a velocity \( V \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Internal Energy**:
The internal energy \( U \) of a gas is primarily the sum of the kinetic energies of the gas molecules. It is a measure of the energy associated with the random motion of the molecules within the gas.

2. **Effect of Box Velocity**:
When the entire box containing the gas is given a velocity \( V \), it is important to note that this motion does not affect the individual velocities of the gas molecules relative to each other. The molecules continue to move with their own velocities, which are determined by their thermal motion.

3. **Relative Motion**:
The kinetic energy of the gas molecules is based on their velocities relative to the center of mass of the gas. Since the box is moving uniformly, the relative velocities of the gas molecules with respect to each other remain unchanged.

4. **Kinetic Energy of Molecules**:
The kinetic energy of the molecules is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of a molecule and \( v \) is its velocity. Since the velocities of the molecules do not change, their kinetic energy remains the same.

5. **Conclusion on Internal Energy**:
Since the internal energy \( U \) is dependent on the kinetic energy of the molecules, and since the kinetic energy of the molecules does not change when the box is given a velocity \( V \), the new internal energy of the gas remains the same as the initial internal energy.

Thus, the new internal energy of the gas is:
\[ U' = U \]

### Final Answer:
The new internal energy of the gas is \( U' = U \).

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