The production of $eta^prime$ mesons in coincidence with forward-going protons has been studied in photon-induced reactions on $^{12}$C and on a liquid hydrogen (LH$_2$) target for incoming photon energies of 1.3-2.6 GeV at the electron accelerator ELSA. The $eta^prime$ mesons have been identified via the $eta^primerightarrow pi^0 pi^0eta rightarrow 6 gamma$ decay registered with the CBELSA/TAPS detector system. Coincident protons have been identified in the MiniTAPS BaF$_2$ array at polar angles of $2^{circ} le theta _{p} le 11^{circ}$. Under these kinematic constraints the $eta^prime$ mesons are produced with relatively low kinetic energy ($approx$ 150 MeV) since the coincident protons take over most of the momentum of the incident-photon beam. For the C-target this allows the determination of the real part of the $eta^prime$-carbon potential at low meson momenta by comparing with collision model calculations of the $eta^prime$ kinetic energy distribution and excitation function. Fitting the latter data for $eta^prime$ mesons going backwards in the center-of-mass system yields a potential depth of V = $-$(44 $pm$ 16(stat)$pm$15(syst)) MeV, consistent with earlier determinations of the potential depth in inclusive measurements for average $eta^prime$ momenta of $approx$ 1.1 GeV/$c$. Within the experimental uncertainties, there is no indication of a momentum dependence of the $eta^prime$-carbon potential. The LH$_2$ data, taken as a reference to check the data analysis and the model calculations, provide differential and integral cross sections in good agreement with previous results for $eta^prime$ photoproduction off the free proton.
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