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In the case where a car is inn circular motion at a roundabout, it moves off in a tangent when its speed surpasses the maximum speed. I understand why such tangential motion happens in the case of a string breaking, but in the case of a car slipping off a roundabout, friction is still acting on the car as it slides. Does the presence of friction not make leaving the circular path not strictly tangential in this case? While the car is slipping off the road, the frictional force will still be influencing it. I'm wondering to what extent this influences the motion of the car.

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Yes, the remaining friction will slightly alter the path, so it is not strictly tangential. But details depend on circumstances (what kind of tires and ground, what speed are the wheels spinning, what direction are they pointing, etc.), so I cannot give you a formula.

BUT: in most cases, sliding friction is a lot smaller than static friction. This means that as soon as you loose contact, the force holding you on the circle drops off very suddenly. For practical purposes it is reasonable to say it is zero.

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