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 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?

### Concept & Kinetic Theory of Gases The pressure exerted by a highly rarefied (low pressure) gas on a moving object is determined by the relative velocity of molecular collisions. The resistive force arises from the difference in momentum transferred to the leading and trailing faces. $$ \Delta P = P_{leading} - P_{trailing} \propto [(u+v)^2 - (u-v)^2] $$ ### Step-by-Step Solution * **Given:** Plate speed $v$, average gas molecule speed $u$ (where $v \ll u$). Constant external force $F$. * **Leading Face Pressure:** Molecules hit the leading face with relative velocity $(u+v)$. The number of collisions per unit time is proportional to $(u+v)$, and the momentum transfer per collision is proportional to $(u+v)$. Thus, leading pressure $P_{leading} \propto (u+v)^2$. * **Trailing Face Pressure:** Molecules hit the trailing face with relative velocity $(u-v)$. Following the same logic, trailing pressure $P_{trailing} \propto (u-v)^2$. * **Pressure Difference (Option A):** $\Delta P = P_{leading} - P_{trailing} \propto [(u+v)^2 - (u-v)^2] = 4uv$. Since mass and number density are constant, $\Delta P \propto uv$. Option A is true. * **Resistive Force (Option B):** The resistive force $F_{res}$ is $\Delta P \times \text{Area}$. Since $\Delta P \propto uv$ and $u$ is the constant average speed of the gas, $F_{res} \propto v$. Option B is true. * **Acceleration (Option C):** Net force $F_{net} = F_{external} - F_{res} = F - kv$ (where $k$ is a constant). Acceleration $a = \frac{F - kv}{m_{plate}}$. As $v$ increases, acceleration decreases. It is not constant. Option C is false. * **Terminal Velocity (Option D):** As the plate speeds up, the resistive force ($kv$) grows until it exactly equals the external force $F$. At this later time, net force is zero, and the plate stops accelerating. Option D is true. ### Exam Strategy & Shortcut In the free-molecular flow regime (low pressure, $v \ll u$), aerodynamic drag is strictly proportional to velocity ($F_{drag} \propto v$). Knowing this standard result immediately validates Option B. If a resistive force scales with velocity against a constant driving force, it will inevitably reach a terminal state where forces balance, instantly validating Option D. ### Common Pitfall Applying standard fluid mechanics drag formulas (where $F_{drag} \propto v^2$) instead of deriving from the kinetic theory of gases. The "very low pressure" condition is the crucial hint to treat this as independent molecular collisions rather than continuous fluid flow. ### Final Answer Therefore, the correct answers are **A, B, and D**.