The analyzing powers for proton elastic scattering ($\vec p A\to pX$) and neutron charge exchange ($\vec n A\to p X$) reactions on nuclei have been measured on $ C$, $CH$, $CH_2$ and $Cu$ targets at incident neutron momenta 3.0 - 4.2 GeV/c by detecting one charged particle in forward direction. The polarized neutron measurements are the first of their kind. The experiment was performed using the Nuclotron accelerator in JINR Dubna, where polarized neutrons and protons were obtained from breakup of a polarized deuteron beam which has a maximum momentum of 13 GeV/c. The polarimeter ALPOM2 was used to obtain the analyzing power dependence on the transverse momentum of the final-state nucleon. These data have been used to estimate the figure of merit of a proposed experiment at Jefferson Laboratory to measure the recoiling neutron polarization in the quasi-elastic $^2H(\vec e,e'\vec n)$ reaction, which yields information on the charge and magnetic elastic form factors of the neutron.
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