Abstract: The transient response of a hanging chain after an initial impact at the free end, is not only an interesting mathematical problem, but also has a certain engineering application. In the theoretical frame of the Lagrange dynamics differential equation, the dynamic response of a hanging chain under the impact of a horizontal impulse-momentum S is analyzed by using the second kind Lagrange's equations under a generalized impulse-momentum, and a general solution is obtained to compute the angular velocity of each segment. This method can be used conveniently to obtain the transient response of a hanging chain with several segments and in different initial conditions.