It is well-known that, in characteristic zero, a rational curve on a smooth surface of non-negative Kodaira dimension is rigid, i.e. do not deform in positive-dimensional families. On the other hand, in positive characteristic, a rational curve on a smooth surface of non-negative Kodaira dimension may not be rigid. However, in this case, the general member of such a family of rational curves is not smooth. In this talk, we show that if a smooth surface of positive characteristic p contains a positive-dimensional family of rational curves such that one member has all delta invariants strictly less than (p-1)/2, then the surface has negative Kodaira dimension. This is a joint work with Tetsushi Ito and Christian Liedtke.