- #31
Harrisonized
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Let θ=0 be the angle at exactly the top of the balloon. If the balloon is expanding uniformly, then none of the points on the balloon can be moving.
Let the ant be not crawling and be at the angle θ=0. Then it is still at angle 0 at any two points in time, and its velocity is zero because it's still 0 distance away from 0.
Now let the ant be not crawling and be at the angle θ=π/4. Then it is still at angle π/4 at any two points in time. However, it's now a linear distance away from θ=0 than it was previously. Therefore, it has a nonzero linear velocity, which violates the fact that we chose its linear velocity to be v=0 for this case, since the ant is not crawling.
Notice that both situations are actually identical, so the ant has both 0 velocity AND it has linear velocity. This is a serious contradiction.
Here's another case. Suppose the ant starts at position 0. It moves to a position 1 unit away from its original rest point and stops moving. However, when it stops moving, its linear displacement is still increasing due to the expansion of the balloon. This violates the fact that we can choose the ant's linear velocity.
The real problem here is that the question isn't well-defined. That's why you found a paradoxical answer. You need an absolute reference frame, and by using angular coordinates exclusively, you've effectively chosen that absolute reference frame to be at the center of the sphere. (This is like choosing the ant to be its own reference frame when its moving.) This is problematic, because then the ant is not in control over its absolute velocity, which we wanted to be true in the problem. (Otherwise, why even bother giving the ant a velocity?)
Forgive me if this sounds a bit rude, but I'm not sure why you asked people for help if you already knew the answer that you seek. I see this problem as a case of modifying the question to correspond to an answer that you want. I guess if you were interested in making test questions you would want to know this, but math is about elucidating concepts, not making them more ambiguous.
Oh and wth does the Doppler effect have anything to do with this?
Let the ant be not crawling and be at the angle θ=0. Then it is still at angle 0 at any two points in time, and its velocity is zero because it's still 0 distance away from 0.
Now let the ant be not crawling and be at the angle θ=π/4. Then it is still at angle π/4 at any two points in time. However, it's now a linear distance away from θ=0 than it was previously. Therefore, it has a nonzero linear velocity, which violates the fact that we chose its linear velocity to be v=0 for this case, since the ant is not crawling.
Notice that both situations are actually identical, so the ant has both 0 velocity AND it has linear velocity. This is a serious contradiction.
Here's another case. Suppose the ant starts at position 0. It moves to a position 1 unit away from its original rest point and stops moving. However, when it stops moving, its linear displacement is still increasing due to the expansion of the balloon. This violates the fact that we can choose the ant's linear velocity.
The real problem here is that the question isn't well-defined. That's why you found a paradoxical answer. You need an absolute reference frame, and by using angular coordinates exclusively, you've effectively chosen that absolute reference frame to be at the center of the sphere. (This is like choosing the ant to be its own reference frame when its moving.) This is problematic, because then the ant is not in control over its absolute velocity, which we wanted to be true in the problem. (Otherwise, why even bother giving the ant a velocity?)
Forgive me if this sounds a bit rude, but I'm not sure why you asked people for help if you already knew the answer that you seek. I see this problem as a case of modifying the question to correspond to an answer that you want. I guess if you were interested in making test questions you would want to know this, but math is about elucidating concepts, not making them more ambiguous.
Oh and wth does the Doppler effect have anything to do with this?
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