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When an car moves forward we go back and we have always read that the reason was that our legs stay in contact with the ground and our body goes back since it was at rest and with sudden motion it still has a tendency to stay at rest . But shouldn’t pseudo force apply too ? I did asked someone that and they said that pseudo force is a result of inertia. If that’s the case how do we say that pseudo force is a result of inertia in the case of centripetal force and in the case of a lift ? I am asking if in a standing bus we go back due to pseudo force or do we go back due to inertia or both ?

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Newtonian mechanics favors the description of mechanical phenomena in inertial reference frames. However, it also allows an equivalent but differently stated description in accelerated reference frames. These two possibilities should never be mixed to avoid confusion and conceptual mistakes.

In the case of a car accelerating forward and the passenger inside, in an inertial reference frame, one describes the effect on the passenger as the effect of the force applied by the accelerating structure of the car on some part of the passenger's body in contact with it. Such force modifies the velocity of the passenger that otherwise would remain constant by inertia.

In the non-inertial reference system of the car, the same effect is due to the presence of a pseudo-force acting on all the bodies in the car in the direction opposite to the direction of acceleration (in the inertial reference frame).