Understanding the math in the Michelson-Morley experiment

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In summary, the textbook excerpt from Modern Physics by Thornton and Rex explains how a swimmer can successfully swim across a swiftly moving river by starting upstream to compensate for the current. The swimmer's velocity throughout the journey is calculated as the square root of their swimming speed squared minus the river's speed squared. The problem being discussed is not clearly defined in terms of velocity versus speed and the reference frames being used. To solve it, one must consider the velocities of the water and the swimmer relative to the bank and use vector components.
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[Mentors' note - No template because this post was moved from the Relativity forum]
Summary:: I'm trying to make sense of the trig/algebra in discussing the experiment. This is an excerpt from the textbook Modern Physics (Thornton/Rex) meant to further explain the set up, however I'm not following. Could someone explain this?

If a swimmer (who can swim with speed v2 in still water) wants to swim across a swiftly moving river (speed v1), the swimmer must start heading upriver, so that when the current carries her downstream, she will move directly across the river. Careful reasoning shows that the swimmer’s velocity is sqrt(v2^2-v1^2) throughout her journey (Problem 4).
 
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The problem is a bit carelessly phrased because it's not respecting the distinction between velocity which is a vector, and speed, which is the magnitude of that vector, and because it's not being clear about which velocities are relative to what.

But with said, you'll be able to get through this problem if you write both the velocity of the water relative to the bank and the velocity of the swimmer relative to the water as vectors with components parallel to and perpendicular to the direction of the water flow. The swimmer's velocity relative to the bank (which is presumably what the problem is lookoing for, although it's not specified) is the sum of these two vectors.
 
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This thread has been moved into the homework forum. You will have to show your work (there's a big hint in the previoius post) and say where you're getting stuck, and then we can help you over the hard spot
 

Related to Understanding the math in the Michelson-Morley experiment

1. How did the Michelson-Morley experiment challenge the concept of the ether?

The Michelson-Morley experiment was designed to measure the speed of light in different directions in order to detect the presence of the ether, a hypothetical substance believed to be the medium through which light traveled. However, the experiment showed no variation in the speed of light, indicating that the ether did not exist.

2. What was the significance of the negative results of the Michelson-Morley experiment?

The negative results of the Michelson-Morley experiment were significant because they directly contradicted the prevailing scientific belief at the time. This led to the development of Albert Einstein's theory of special relativity, which states that the laws of physics are the same for all observers in uniform motion, regardless of their relative velocity.

3. How did the Michelson-Morley experiment pave the way for future scientific discoveries?

The negative results of the Michelson-Morley experiment challenged the existing understanding of light and motion, ultimately leading to the development of new theories and discoveries in the field of physics. This experiment paved the way for further research and advancements in fields such as relativity, quantum mechanics, and the study of the fundamental properties of the universe.

4. Did the Michelson-Morley experiment have any practical applications?

While the Michelson-Morley experiment did not have any direct practical applications, it had a significant impact on our understanding of the fundamental nature of the universe. It also laid the foundation for the development of new technologies, such as the GPS system, which relies on the principles of relativity.

5. How does the Michelson-Morley experiment continue to be relevant in modern physics?

The Michelson-Morley experiment continues to be relevant in modern physics as it serves as a cornerstone in the development of special relativity and the understanding of the nature of light and motion. It is also often referenced in discussions about the nature of the universe and the search for a unified theory of physics.

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