A flat plate is moving normal to its plane through a gas under the...

${v}_{1} = u + 2v$

${v}_{2} = \left(u - 2v\right)$

$\Delta{v}_{1} = \left(2u + 2v\right)$

$\Delta{v}_{2} = \left(2u - 2v\right)$

${F}_{1} = \dfrac{d{p}_{1}}{dt} = ρA\left(u + v\right)\left(2u + 2v\right)$

${F}_{2} = \dfrac{d{p}_{2}}{dt} = ρA\left(u - v\right)\left(2u - 2v\right)$

$= 2ρA{\left(u + v\right)}^{2}$

$= 2ρA{\left(u - v\right)}^{2}$

$\DeltaF = {F}_{1} - {F}_{2}$ ($\DeltaF$net force due to the air molecules on the plate)

$= 2ρA\left(4uv\right) = 8ρAuv$

$P = \dfrac{\DeltaF}{A} = 8ρ\left(uv\right)$

${F}_{net} = \left(F - \DeltaF\right) = ma$ (m is mass of the plate)

$F - \left(8ρAu\right)v = ma$