A flat plate is moving normal to its plane through a gas under the action of a constant force F. The gas is kept at a very low pressure. The speed of the plate v is much less than the average speed u of the gas molecules. Which of the following options is/are true ?

Options

A. The pressure difference between the leading and trailing faces of the plate is proportional to
B. The plate will continue to move with constant non-zero acceleration, at all times
C. At a later time external force F balances the resistive force
D. At a later time external force F balances the resistive force

Explanation

To solve the problem, we need to analyze the motion of a flat plate moving through a gas at low pressure, under the influence of a constant force \( F \). The speed of the plate \( v \) is much less than the average speed \( u \) of the gas molecules. We will examine the implications of this scenario step-by-step.

### Step-by-Step Solution:

1. **Understanding the System**:
- A flat plate is moving through a gas. The gas is at very low pressure, meaning the density of gas molecules is low.
- The plate experiences a constant force \( F \) and moves with speed \( v \), which is much smaller than the average speed \( u \) of the gas molecules.

2. **Pressure Difference**:
- As the plate moves, it interacts with gas molecules. The leading face of the plate will experience a higher pressure due to the incoming gas molecules, while the trailing face will experience a lower pressure as the gas molecules move away.
- The pressure difference \( \Delta P \) between the leading and trailing faces of the plate can be expressed in terms of the average speed of the gas molecules \( u \) and the speed of the plate \( v \).
- From kinetic theory, the pressure difference is proportional to the product of the average speed of the gas molecules and the speed of the plate:
\[ \Delta P \propto u \cdot v \]

3. **Resistive Force**:
- The resistive force \( F_r \) acting on the plate due to the gas molecules can be derived from the pressure difference:
\[ F_r = \Delta P \cdot A \]
- Substituting the expression for \( \Delta P \):
\[ F_r \propto A \cdot (u \cdot v) \]
- This indicates that the resistive force is proportional to the product of the average speed of the gas molecules and the speed of the plate.

4. **Constant Acceleration**:
- Initially, the plate may accelerate due to the net force acting on it. However, as it continues to move, the resistive force will increase until it balances the applied force \( F \).
- At some later time, the external force \( F \) will balance the resistive force \( F_r \), leading to constant velocity (zero acceleration).

5. **Conclusion**:
- The pressure difference between the leading and trailing faces of the plate is proportional to \( u \cdot v \).
- At a later time, the external force \( F \) will balance the resistive force, resulting in no net acceleration.

### Final Statements:
- The correct options based on the analysis are:
- The pressure difference between the leading and trailing faces of the plate is proportional to \( u \cdot v \).
- At a later time, the external force \( F \) balances the resistive force.