Question

Michelson and Morley used an interferometer to show that the speed of light is constant, regardless of Earth's motion through any perceived luminiferous aether. An analogy can be understood from the different times it takes for a rowboat to travel two different round-trip paths in a river that flows at a constant velocity \((u)\) downstream. Let one path be for a distance \(D\) directly across the river, then back again; and let the other path be the same distance \(D\) directly upstream, then back again. Assume that the rowboat travels at constant speed, \(v\) (with respect to the water), for both trips. Neglect the time it takes for the rowboat to turn around. Find the ratio of the cross-stream time divided by the upstream-downstream time, as a function of the given constants.

Short answer

Answer: The ratio is given by the formula: \(ratio = \frac{2(v^2 - u^2)}{(v - u)(v + u)}\)

Step-by-step solution

Calculate cross-stream time

First, let's find the time it takes for the rowboat to travel directly across the river and back (cross-stream path). The total distance traveled is \(2D\). The rowboat's velocity in relation to the water is \(v\), but it must also factor in the river velocity \(u\). To do this, we can use the Pythagorean theorem to find the boat's effective velocity: \(v_{eff} = \sqrt{v^2 - u^2}\) Then, we can find the time it takes: \(t_{cross} = \frac{2D}{v_{eff}}\)

Calculate upstream-downstream time

Now, let's find the time it takes for the rowboat to travel directly upstream and then back downstream (upstream-downstream path). The total distance traveled is also \(2D\). However, the boat's velocity is now affected by the river's velocity differently: When going upstream, the effective velocity is: \(v_{up} = v - u\) The time it takes to travel upstream is: \(t_{up} = \frac{D}{v_{up}}\) When going downstream, the effective velocity is: \(v_{down} = v + u\) The time it takes to travel downstream is: \(t_{down} = \frac{D}{v_{down}}\) The total time of the upstream-downstream path is: \(t_{up-down} = t_{up} + t_{down}\)

Calculate the ratio of cross-stream time and upstream-downstream time

Finally, we can find the ratio of the cross-stream time divided by the upstream-downstream time: \(ratio = \frac{t_{cross}}{t_{up-down}}\) Plug in our expressions for \(t_{cross}\), \(t_{up}\), and \(t_{down}\) into the ratio: \(ratio = \frac{\frac{2D}{\sqrt{v^2 - u^2}}}{\frac{D}{v - u} + \frac{D}{v + u}}\) Simplify the expression: \(ratio = \frac{2(v^2 - u^2)}{2D(v^2 - u^2)}\) The final ratio as a function of the given constants is: \(ratio = \frac{2(v^2 - u^2)}{(v - u)(v + u)}\)

Key concepts

Interferometer

An interferometer is a device that splits a beam of light into two paths and then recombines them. By doing this, it can measure very small distances or changes in distance by looking at interference patterns. Think of it like a super precise ruler for light! This device was crucial in the Michelson-Morley experiment. Michelson and Morley used it to detect any differences in the speed of light as the Earth moved through what was once thought to be 'luminiferous aether'.
In an interferometer:

• Light is divided by a beam splitter.
• The separate light beams travel different paths.
• They are recombined to create an interference pattern.

These interference patterns shift based on changes in the path lengths, allowing scientists to measure extremely small differences in distance or time.

Speed of Light

The speed of light, denoted as 'c', is a fundamental constant in physics. It's the speed at which light travels in a vacuum and is approximately 299,792,458 meters per second. In the Michelson-Morley experiment, it was assumed that if Earth moved through the supposed 'aether', the speed of light would change in different directions due to this motion. One unique property of the speed of light is its constancy in a vacuum, regardless of the motion of the source or observer. This groundbreaking realization laid the foundation for Einstein's theory of relativity. The experiments showed that no matter how Earth moved, the speed of light remained the same in all directions, suggesting there was no 'aether' affecting its velocity.

Luminiferous Aether

In the 19th century, the luminiferous aether was hypothesized to be the medium through which light waves traveled, much like air helps sound waves propagate. Scientists believed this invisible substance filled all space and facilitated the transmission of light. The idea was that much like a boat experiences water resistance, light would experience changes in speed if Earth moved through this 'aether'.
However, the Michelson-Morley experiment, using the interferometer, found no difference in the speed of light in various directions. This effectively debunked the idea of the 'aether'. The absence of evidence for aether paved the way for modern physics, leading to the acceptance that light does not require a medium to travel through space.

Velocity Calculation

When calculating velocity in the presence of a flow or current, like a river or wind, it's essential to understand relative velocity. In the Michelson-Morley experiment analogy:

• A rowboat moving in a river faces the water's current, just as Earth supposedly faced the 'aether'.
• The velocity of the boat changes whether it's moving with or against the current.

To find the effective velocity:

• For crossing a stream, Pythagoras' theorem helps find a resultant velocity.
• Upstream velocity is factored by subtracting the current.
• Downstream velocity adds the influence of the current.

The Michelson-Morley experiment used these types of relative velocity calculations to determine if light speed changed due to Earth's motion through 'aether'. Their results showed no change, indicating light's velocity is unaffected by such motion.