Inpatient admission for the psychological aftermath of injury is not common. The human being is quite resilient and while back, neck and other painful orthopedic injuries are painful and distressing, they do not typically create the degree of emotional decompensation that necessitates admission to a mental health unit.
However, there are exceptions. Some of the more severely traumatic experiences such as horrific motor vehicle accidents or assaults can rise to the level where the person is not able to compensate as an outpatient. At that time, hospitalization should not be postponed.
There is individual variability to this even within the more profound accidents. While one individual may be able to cope with a severely violent trauma, others are overwhelmed by being in the proximity of such a situation, even if they were not personally involved/injured. There are those who appear for all purposes to be almost immune to stress and calmly/objectively address situations as they occur. And there are others for whom the emotional trauma does not emerge for weeks or months after the horrific event.
In general, the more violent the situation, and the more destructive to body integrity and emotional safety, the more likely to trigger a sense of being overwhelmed from which the patient cannot rebound in their home environment and must have a more secure hospital setting.