This study investigates selected aspects of the dynamic behaviour of mine hoists during the emergency braking in an event of overtravel. Characteristics of the braking force that needs to be applied in the headgear and in the pit bottom to arrest the conveyance in the event of an overtravel are derived from laboratory and industrial test data and recalling the results reported in literature. The real hoist installation is replaced by a model whereby the equations of motion of rope elements are written as for elastic strings, taking into account the variable length of the hoisting rope section between the Koepe pulley and the conveyance being arrested in the head tower. Analytical formulas are provided whereby the displacement of the top conveyance with the payload for the constant elasticity coefficient of the hoisting rope section between the conveyance being arrested in the head tower and the Koepe pulley is expressed as the function of the braking force and of the operational parameters of the hoist gear. The hoist operation is investigated in the event of emergency braking, taking into account the two aspects of the cycle: - the time required for the conveyance to be stopped, - the distance travelled by the conveyance until it is stopped. The results of the dynamic analysis of the hoist installation in the conditions of emergency braking may be utilised in selection of the effective and adequate braking system guaranteeing the safety of the system operation.