Would the trachea collapse when you were exhaling if there were no rings of cartilage in its walls

The answer is YES. The trachea would collapse during exhaling due to the negative pressure developed as a result of evacuating air from the lungs. Physics dictates that the area with high pressure (outside environment) pressure on the region with low air pressure (in the lungs during exhalation).

Yes, the trachea would collapse in this situation.

The trachea is a tube composed of cartilage which connects the pharynx and larynx to the lungs. It is a critical structure for all air-breathing animals and its main responsibility is to facilitate the passage of air. The trachea is surrounded by several c-shaped and incomplete cartilaginous rings which prevent it from collapsing.