  Using $(10.087\pm0.044)\times10^{9}$ $J/\psi$ events collected with the
BESIII detector at the BEPCII storage ring, the processes $\Lambda
p\rightarrow\Lambda p$ and $\bar{\Lambda}p\rightarrow\bar{\Lambda}p$ are
studied, where the $\Lambda/\bar{\Lambda}$ baryons are produced in the process
$J/\psi\rightarrow\Lambda\bar{\Lambda}$ and the protons are the hydrogen nuclei
in the cooling oil of the beam pipe. Clear signals are observed for the two
reactions. The cross sections in
$-0.9\leq\rm{cos}\theta_{\Lambda/\bar{\Lambda}}\leq0.9$ are measured to be
$\sigma(\Lambda p\rightarrow\Lambda
p)=(12.2\pm1.6_{\rm{stat}}\pm1.1_{\rm{sys}})$ mb and $\sigma(\bar{\Lambda}
p\rightarrow\bar{\Lambda} p)=(17.5\pm2.1_{\rm{stat}}\pm1.6_{\rm{sys}})$ mb at
the $\Lambda/\bar{\Lambda}$ momentum of $1.074$ GeV/$c$ within a range of
$\pm0.017$ GeV/$c$, where the $\theta_{\Lambda/\bar{\Lambda}}$ are the
scattering angles of the $\Lambda/\bar{\Lambda}$ in the $\Lambda
p/\bar{\Lambda}p$ rest frames. Furthermore, the differential cross sections of
the two reactions are also measured, where there is a slight tendency of
forward scattering for $\Lambda p\rightarrow\Lambda p$, and a strong forward
peak for $\bar{\Lambda}p\rightarrow\bar{\Lambda}p$. We present an approach to
extract the total elastic cross sections by extrapolation. The study of
$\bar{\Lambda}p\rightarrow\bar{\Lambda}p$ represents the first study of
antihyperon-nucleon scattering, and these new measurements will serve as
important inputs for the theoretical understanding of the (anti)hyperon-nucleon
interaction.
