The fact that a bungee jumper can reach an acceleration greater than the acceleration of gravity is, also from a physics point of view, intriguing. Taking only gravity into account, it can be explained by applying conservation of energy or by deriving carefully the equation of motion in a Newtonian approach. In this article we show how it can be done straightforwardly via the Lagrangian approach. We will also apply this method to the fall of a block with a chain hanging underneath and touching the ground floor, and to the fall of a chain on a scale. These examples of systems of variable mass may motivate physics students to learn and use the Lagrangian approach, and may let them appreciate the benefits of this approach in some physics problems compared to Newtonian mechanics, with which they are already familiar.