= Solution
The collision flux contains an extra normal <velocity> $v\cos\theta$: a faster atom sweeps out a longer approach cylinder per unit time. Integrating the <Maxwell-Boltzmann distribution> over the incoming hemisphere gives
$$
J=nA\int_0^\infty v^3e^{-\alpha v^2}dv\int_0^{\pi/2}\sin\theta\cos\theta\,d\theta\int_0^{2\pi}d\phi=nA\pi I_3.
$$
Therefore
$$
\boxed{J=n\sqrt{\frac{k_BT}{2\pi m}}=\frac14n\overline v.}
$$
Here $J$ is the number striking unit area per unit time and $\overline v=\sqrt{8k_BT/(\pi m)}$ is the bulk mean speed. A simple factor of one half would count the incoming hemisphere but miss the projection factor and velocity weighting.
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