$\alpha$-cluster knockout from ${\mathrm{Be}}^9$, ${\mathrm{C}}^{12}$, and ${\mathrm{O}}^{16}$ by 160-MeV protons was investigated. The outgoing proton and $\alpha$ particle were detected in coincidence and their momenta measured with good resolution. Knockout, sequential decay, and spallation reactions were observed. Proton and $\alpha$ detection angles were varied so as to test the plane-wave impulse-approximation (PWIA) model of the knockout reaction over a wide range of the effective $p$-cluster interaction. The method, apparatus, and carbon results are discussed in this paper. Measured values of the fully differential cross section ($\frac{d\sigma }{d{\Omega }_{\alpha }d{\Omega }_{p}d{T}_p}$) are given, and are shown to vary nearly as predicted by the PWIA. The errors due to finite detector sizes (finite momentum resolution) are discussed in some detail. The recoil correction to the usual PWIA expression for the cross section is derived. The branching ratio to the first excited state of the residual ${\mathrm{Be}}^8$ nucleus, as well as the effective number of $\alpha$ clusters in ${\mathrm{C}}^{12}$, are discussed and compared with theoretical predictions.

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