In the 1800s scientists understood that light acted as a wave. Because sound waves travel through air, and tidal waves travel in water, they believed light waves travelled through a medium as well, what they called the aether. The theory implied that light from the sun had to “swim” in the aether to reach the earth. And just as a strong wind affects the propagation of sound, the presence of an aether wind would affect the flow of light. But how could one detect the aether and the speed of the aether wind?

In 1887, two scientists developed an apparatus to detect the aether wind. One of the scientists used a math puzzle to explain the principle behind the experiment. So consider the following problem.

Two equally strong swimmers race at a river. They start and end at the same point, but they take different paths. Swimmer 1 goes to the closest point across the river, 100 feet in width, and returns. Swimmer 2 goes directly upstream (against the current) for 100 feet and then returns by swimming downstream. Who wins the race? Assume the river flows at 3 feet per second, and the swimmers have a speed of 5 feet per second.

Note that both swimmers are influenced by the flow of the river. Swimmer 1 has to fight the flow of the river to make a straight line path across and back. Swimmer 2 is actively swimming against the river flow in one direction and then swimming with the river flow on the return trip.

The solution to the puzzle, and its relation to the aether winds, is presented after the jump.

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Answer to The Most Famous Failed Experiment Math Problem

The easier case is the swimmer 2, who swims upstream (against the current) in one direction and then downstream (with the current) in the other. The net speed is equal to the combined effect of the swimmer’s speed and the river’s flow.

Going downstream the swimmer’s speed is the sum of his speed plus the river’s flow (8 = 5 + 3), and going upstream his speed is his rate minus the river’s flow (2 = 5 – 3). The time downstream is 100/8 = 12.5 seconds and the time upstream is 100/2 = 50 seconds. The total time is therefore 62.5 seconds.

Swimmer 1 is trickier because he is swimming across the flow of the river. The path he swims will be the net result of his velocity and the river flow’s velocity. We can think about this in terms of vector math: the swimmer’s vector plus the river’s vector must be equal to a straight line vector.

In the diagram, the swimmer’s vector is the hypotenuse and the river flow’s is the short leg. The numbers have been chosen with a purpose: the swimmer’s vector has a speed of 5 feet per second, the river flow’s is 3 feet per second, and the vectors form a right triangle. Therefore, we must have a 3-4-5 right triangle! The swimmer’s net speed is 4 feet per second. Note the size of the vectors will be the same for the trip across the river and the return trip, so the swimmer’s speed is 4 feet per second on each trip.

Swimmer 1 takes 100/4 = 25 seconds for each of the two trips. Therefore the cross-stream swimmer takes 50 seconds in all, which is shorter than the 62.5 seconds of the upstream swimmer.

The answer is swimmer 1 wins. And this turns out always to be true: swimmer 1, the cross-stream swimmer, always wins (so long as their swimming speed is faster than the flow of the river).

The Michelson-Morley experiment

In 1887 the scientists Michelson and Morley developed an apparatus to detect the aether wind. The device was called an inferometer, and it worked on the same principle as the swimming puzzle. (Michelson devised the puzzle to explain it to his children).

The inferometer used a set of mirrors to split up a beam of light into two beams. The two beams then “raced” just like the two swimmers. One beam of light was like the cross-stream swimmer that was sent to the closest point across a certain distance. The other beam was like the upstream swimmer and it was sent the same distance against the aether wind. If the aether wind existed, then the light beam that “swam” cross-stream would win every time.

Animation from Wikipedia, CC BY-SA 3.0.

In the above animation, the red dot is like the cross-stream swimmer and the blue dot is the upstream swimmer. If the aether wind existed, the red dot would travel faster like the animation on the right. What happened was the two dots arrived at about the same time, like what is depicted on the left animation.

The experiment provided strong evidence that the aether wind did not exist. This is celebrated as one of the most famous experiments in physics.

Source of puzzle and further reading

The Michelson-Morley Experiment, Michael Fowler U. Va. Physics 9/15/08.

http://galileo.phys.virginia.edu/classes/109N/lectures/michelson.html