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It can fly in almost any direction.

But the first thing you need to decide is what "frame of reference" you want to use when considering this question. If one says the mud "flies backward" does this mean that a person sitting in the car sees it fly backward or that a person standing on the road sees it fly backward? If the car is moving, then these are NOT the same.

For example, if the car is traveling 40 mph, and a person in the car sees the mud fly backward at 25 mph, a person standing on the road will see the same mud going in the same direction as the car at 15 mph.

Assume that the tires of the car rolling along the ground. That is you are NOT spinning your tires in the mud. If the car is stationary spinning it's wheels, then the following answers would have to be adjusted a bit.

Keep in mind here that it is incorrect to think of the tires as "throwing" the mud. It's really the tire tread moving away from the mud as the tread moves in a circle and the mud continues in a straight line thanks to its momentum.

From the point of view of someone in the car, the mud will be seen to fly forward, backward, up, and to some extent to the sides. The very top of the tire is moving at twice the speed of the car, so if the mud coming off here is going faster than the car and flys forward of the car. The bottom of the tire (where it touches the ground) is moving at zero mph. When the mud comes off here (or just a little later as the tire tread starts to lift from the ground), the mud will be going slower than the car so it will fly backward relative to the car. Likewise mud coming off the rear most part of the tire will fly up. And mud coming off the very front will fly down (but hit the ground and not be seen).

Now from the point of view of a person standing on the side of the road: Basically all the mud they see flying will be flying up and in the same direction as the car. The slowest part of the vehicle is where the tire touches the ground. This spot on the tire is going zero mph. So, and mud that comes off here will be standing still relative to the person on the side of the road. But, it will also be sitting in a blob on the pavement, so it would not be "flying". Every other point on the tires and car has at least some velocity in the same direction that the car is traveling. So, all other mud that comes off will be moving in the same direction as the car. But much of it will also be moving up or down. The part moving down quickly hits the pavement so all the person beside the road really sees is mud flying up and "following" the car.

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Q: When mud flies off the tire of a moving car in what direction it fly?
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