Final answer:
The vestibule, specifically the utricle and saccule, contain otoliths which detect linear acceleration through shearing forces that bend the stereocilia, generating electrical signals sent to the brain.
Step-by-step explanation:
The organ that detects linear acceleration is the vestibule, specifically the utricle and saccule within the vestibule which contain the maculae. These sensory structures include otoliths, which are small crystals that lay on top of the gelatinous layer within the utricle and saccule. When the head moves linearly—as in your instance, when a car suddenly decelerates—the difference in inertia between the hair cell stereocilia and the otolithic membrane leads to shearing forces that cause the stereocilia to bend in the direction of the acceleration. This bending is transduced into electrical signals, which are then sent to the brain to interpret the movement.