Question

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?

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

Answer 1

Answer: (A), (B), (D)

At a later time the external force $F$ balances the resistive force
The resistive force experienced by the plate is proportional to $v$
The pressure difference between the leading and trailing faces of the plate is proportional to $uv$

$v_1=u+2v$

$\Delta V_1=(2u+2v)$

$F_1=\frac{d{p}_1}{dt}=\rho A(u+v)\times (2u+2v)=2\rho A(u+v{)}^2$

$v_2=(u-2v)$

$\Delta v_2=2u-2v$

$F_2=\frac{d{p}_2}{dt}=\rho A(u-v)\times (2u-2v)=2\rho A(u-v{)}^2$

As $F_1$ and $F_2$ are acting in opposite direction so

$F_{net}=F_1-F-2=2\rho A\left(4uv\right)=8\rho Avu$

$P=\frac{\Delta F}{A}=8\rho uv$

$F_{net}=(F-\Delta F)=ma$ m is the mass of the plate.

$F-\left(8\rho Auv\right)=ma$