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Explain why a passenger on a bus falls backwards when the bus moves forwards and falls forward when the bus stops?
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Equal and opposite forces. Think of it as similar to how the force of gravity is cancelled by the normal force.
I think the for this is due to the law of inertia. When the bus is moving the lower part of the body in adjacent contact with the bus also moves, on applying brakes the upper part moves forward due to its law of inertia where as the lower part in contact with the bus stops as corresponding to the movement of the bus. I think this is the reason for a person to fall forward or backward in a bus on application of the brakes.
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