This thesis describes how to evaluate a first-order approximation of the vibration induced on a beam that is vertically curved and experiences a moving load of non-constant velocity. The curved beam is applicable in the example of a roller coaster. The present research in the field does not consider a curved beam nor can similar research be applied to such a beam. The complexity of the vibration of a curved beam lies primarily in the description of the variable magnitude of the moving load applied. Furthermore, this motion is also variable. This thesis will present how this beam will displace in response to the moving load. The model presented can be easily manipulated as it considers most variables to be functions of time or space. The model will be compared to existing research on linear beams to ensure the unique response of a curved beam.