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?

Step 1: Analyze the pressure on the leading and trailing faces.

In the reference frame of the plate, the gas molecules strike the leading face with a relative speed proportional to $u + v$, and the trailing face with a relative speed proportional to $u - v$.

$$P_{\text{lead}} \propto (u+v)^2$$ $$P_{\text{trail}} \propto (u-v)^2$$

Step 2: Calculate the pressure difference.

Subtract the pressure on the trailing face from the pressure on the leading face.

$$\Delta P \propto (u+v)^2 - (u-v)^2$$ $$\Delta P \propto 4uv$$ $$\Delta P \propto uv$$

This shows the pressure difference is proportional to $uv$, making Option (1) true.

Step 3: Determine the resistive force on the plate.

The resistive force $R$ is the product of the pressure difference and the plate area $A$.

$$R = \Delta P \cdot A$$

Since $u$ and $A$ are constants, we obtain the relation for the resistive force.

$$R \propto v$$

This shows the resistive force is proportional to $v$, making Option (2) true.

Step 4: Analyze the acceleration of the plate.

Apply Newton's second law to write the equation of motion, where $\beta$ is a constant of proportionality.

$$m\frac{dv}{dt} = F - \beta v$$

Solving this differential equation with $v(0) = 0$ yields the velocity as a function of time.

$$v(t) = \frac{F}{\beta}\left(1 - e^{-\beta t/m}\right)$$

Since the acceleration decreases exponentially over time, Option (3) is false.

Step 5: Evaluate the terminal state of the plate.

As $t \to \infty$, the velocity approaches a constant terminal value.

$$v \to \frac{F}{\beta}$$

At this terminal speed, the net force becomes zero.

$$F = \beta v = R$$

Thus, the external force $F$ eventually balances the resistive force, making Option (4) is true.

Answer: The true options are (1), (2), and (4).