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Math Help - Getting Nowheres Fast

  1. #1
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    Getting Nowheres Fast

    As a preliminary, think about the old phonograph records spinning. The closer to the center you get, the slower the speed as you approach the center and there's no motion at the exact center.

    Now for my problem. Assume you have a car with perfectly rounded tires moving at 25 mph. Let's focus on the perimeter of a tire and see what happens when you're following a point along the circumference starting from the top of the tire.

    From the very top is when the tire is moving at its fastest relative to the ground. As the car is moving forward, the point along the circumference is slowing down as it's approaching the ground and when it's touching the ground, the tire isn't moving at all relative to the ground.

    So we have a paradox. The car is moving forward at 25 mph, yet its four tires aren't moving at all at the point they're touching the ground. Can you find a flaw in the logic or can you resolve this paradox?
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  2. #2
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    I'm guessing an optical illusion caused by moving you line of vision along the ground with the wheel?
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  3. #3
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    Well, with your logic, aren't you also saying that the tire is not moving when the point is at the top of the tire?
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  4. #4
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    rrm74001 response

    Did you read?, "From the very top is when the tire is moving at its fastest relative to the ground."
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  5. #5
    MHF Contributor Bruno J.'s Avatar
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    There are two different speeds to take into account. There is the forward speed of the car relative to the ground, and the speed of the point on the tire relative to the centre of the tire.
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  6. #6
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    Thank you Bruno J.

    "...and the speed of the point on the tire relative to the centre of the tire."

    In my problem I'm focused on the circumference of the tire and the way it moves relative to the ground. It's clear that where the tire is touching the ground, the tire isn't moving at all no matter how fast the car is moving.
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  7. #7
    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by wonderboy1953 View Post
    It's clear that where the tire is touching the ground, the tire isn't moving at all no matter how fast the car is moving.
    Why is that clear?
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  8. #8
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    Response

    "Why is that clear?"

    Reread my third paragraph in the top post.
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  9. #9
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    Better response (Bruno J)

    Relative to the tire's center, any point along the circumference is moving at 25 mph as given in the problem. The point at the top of the tire is moving forward at 25 mph relative to the center, the point at the tire's bottom along the perfectly flat ground is also moving at 25 mph, but moving backwards relative to the center. The ground is also moving 25 mph backwards relative to the tire's center (as given in the problem). Therefore since both the bottom of the tire and the ground are moving backwards, relative to the tire's center, at 25 mph, then this is the same as the four tires not moving at all relative to the ground at the point they're touching while the car with the four central tire points are moving forward at 25 mph.

    I hope this clears up any illusions.
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  10. #10
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    Resolution (hopefully)

    Just because the points where the tires touch the ground aren't moving doesn't mean that the rest of the tire can't move. Also the non-moving ground touching points turn into moving points at another point in time so I think this clears up the paradox.
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  11. #11
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    Tire Speed, Top vs. Bottom

    I rushed out with a pail of water and washed my tires carefully.....
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  12. #12
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    Not bad Wilmer

    I'm glad to see someone else besides me thinking about these things.
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