Solid surfaces, in particular polymer surfaces, are able to adapt upon contact with a liquid. Adaptation results in an increase in contact angle hysteresis and influences the mobility of sliding drops on surfaces. To study adaptation and its kinetics, we synthesized a random copolymer composed of styrene and 11-25 mol% acrylic acid (PS/PAA). We measured the dynamic advancing (theta(A)) and receding (theta(R)) contact angles of water drops sliding down a tilted plate coated with this polymer. We measured theta(A) approximate to 87 degrees for velocities of the contact line <20 mu m/s. At higher velocities, theta(A) gradually increased to similar to 98 degrees. This value is similar to theta(A) of a pure polystyrene (PS) film, which we studied for comparison. We associate the gradual increase in theta(A) to the adaptation process to water: The presence of water leads to swelling and/or an enrichment of acid groups at the water/polymer interface. By applying the latest adaptation theory (Butt et al. Langmuir 2018, 34, 11292), we estimated the time constant of this adaptation process to be << 1 s. For sliding water drops, theta(R) is similar to 10 degrees lower compared to the reference PS surface for all tested velocities. Thus, at the receding side of a sliding drop, the surface is already enriched by acid groups. For a water drop with a width of 5 mm, the increase in contact angle hysteresis corresponds to an increase in capillary force in the range of 45-60 mu N, depending on sliding velocity.