The atomistic mechanism of dissociative adsorption of ethylene molecules on a Ni cluster is investigated by ab initio molecular-dynamics simulations. The activation free energy to dehydrogenate an ethylene molecule on the Ni cluster and the corresponding reaction rate is estimated. A remarkable finding is that the adsorption energy of ethylene molecules on the Ni cluster is considerably larger than the activation free energy, which explains why the actual reaction rate is faster than the value estimated based on only the activation free energy. It is also found from the dynamic simulations that hydrogen molecules and an ethane molecule are formed from the dissociated hydrogen atoms, whereas some exist as single atoms on the surface or in the interior of the Ni cluster. On the other hand, the dissociation of the C-C bonds of ethylene molecules is not observed. On the basis of these simulation results, the nature of the initial stage of carbon nanotube growth is discussed.