Thanks for this interesting question, which relates with the safety while flying. Although there are many safety points, which passengers should keep in mind during the flight, this is one of the common point, passengers most often neglect and in some cases, I have seen (during my regular flight duties) many times, passengers get injured.
car moves in forward direction with high speed. If it stops suddenly or slowsdown suddenly, we tend to move forward and may hit the front seat (for rear passenger) or steering (Driver). Even if the car is moving with 60 km/hr, and collides with opposite car coming with 40 km per hour, the collision impact is felt with the addition of these two speed (relative velocity). i.e. the driver will feel as if hit at 100 km per hour for both the car persons. The body which was moving with 60 km/hr feels to move ahead with 100 km /hr, when it collides and the car is now standstill after collision. That means the body will be thrown forward with 100 km/hour and hit the steering (front seat for rear passenger).
As such body is restricted to move ahead for safety during such probable fatal collision. and so the body is tied to restrict moving ahead suddenly.
In aircraft, there is no chance of sudden brakes in air, slowing speed suddenly or little chance (seldom) of air collision (If happens its sure to death irrespective of any belt) as such the front movement is controlled. Therefore the body belt thru shoulder is not required.
However, when we fly in air pockets (the coloumn of air moving up or down) the aircraft moves up or down suddenly. When the aircraft passes through a down going air pockets, it is sucked down suddenly and the aircraft descends (sucked down) even more than a rate of free falling object i.e. more than ‘g’(acceleration due to gravity). some times double of gravity ie at 19.6 m/sec square. The passengers inside the aircraft do also fall but with acceleration due to gravity ie. at the rate of 9.8m/sec square acceleration.
when the aircraft falls quicker than the passengers inside, the passengers remain up but the seat and aircraft go down. In such situation, the passenger feels lifted up and hit the ceiling of the cabin suddenly which hurts his/her head.
courtesy: google
While flying, the up and down movement of the aircraft may cause injury to a passenger, if his up and down movement is not restricted.
above is one of the picture, where a cabin crew is hitting the ceiling while serving in turbulence passing flight.
Out side, even in clear weather, the air pockets are not seen, but columns of such air currents may be active. To avoid up and down movement in flying which may invite head injury, the seat belt is mandatory during turbulent passing flights. and so, seat belt signs are kept on.
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