An oscillator of two lumped masses linked through a vertical spring moves forward in the horizontal direction, initially at a certain height, over a horizontal Euler beam and descends on it due to its own weight. Vibration of the beam and the oscillator is excited at the onset of the ensuing impact. The impact produced by the descending oscillator is assumed to be either perfectly elastic or perfectly plastic. If the impact is perfectly elastic, the oscillator bounces off and hits the beam a number of times as it moves forward in the longitudinal direction of the beam, exchanging its dynamics with that of the beam. If the impact is perfectly plastic, the oscillator (initially) sticks to the beam after its first impact and then may separate and reattach to the beam as it moves along the beam. Further events of separation and reattachment may follow. This interesting and seemingly simple dynamic problem actually displays rather complicated dynamic behaviour and has never been studied in the past. It is found through simulated numerical examples that multiple events of separation and impact can take place for both perfectly elastic impact and perfectly plastic impact (though more of these in the case of perfectly elastic impact) and the dynamic response of the oscillator and the beam looks noisy when there is an event of impact because impact excites higher-frequency components. For the perfectly plastic impact, the oscillator can experience multiple events of consecutive separation from the beam and subsequent reattachment to it.