A Tale of Two Tugboats

Two tugboat captains are sitting in a bar arguing about who has the more powerful boat. They decide to have a contest to settle the dispute so set out into the estuary with a hawser between the boats having agreed with the operators of two local radar stations that they would monitor the test. They run the engines up to full power and after pulling for some time one sailor realises that he has been working against the tide all the time. "If I've held position against his boat and the tide, I'll win for sure as soon as the tide turns.", he thinks. A little later, the tide turns. After another half hour getting nowhere, the captain calls his opponent and they agree to weigh anchor and go back to the pub.

When the get to the pub, they are surprised to find the radar operators arguing over how far apart the tugs were during the trial. The operator from Site A is saying that he had checked the ranges carefully and the boats were 400m apart in range. The operator from Site B meanwhile is insisting that the difference in range was only 300m and it was the lateral separation that was 400m since he had measured it as 4 degrees and the overall range was about 5.73km. "'No, no'; says the first operator, "the lateral separation was definitely 300m, it was the radial separation that was 400m."

At this point, in walks a physicist looking for a relatively cool beer. He points out that actually they are both right since the radar sites are at different locations. By using Pythagoras' Theorem, he shows that the reality they are both describing is that the hawser between the boats was 500m.

The sailors and radar operators then see the error of their ways and agree that the best solution is to throw the physicist in the harbour - nobody likes a smartass!

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Here is a view of the tugs and the radar sites: Picture of Tugs and radar sites

The radar operators readings were:

Operator: Radar A Radar B
Range: 5730m 5730m
Angular Separation (deg): 3 4
Lateral Separation: 300m 400m
Radial Separation: 400m 300m
sqrt(Lateral2 + Radial2): 500m 500m

Both the angular (hence lateral) separation and the radial separation (shown by the short perpendiculars near the boats) vary according to the position of the radar site. However, if we calculate the quantity:

sqrt(Lateral2 + Radial2)

we always get the value 500m. There should be nothing surprising about this, its just the length of hawser.

Now lets look at a similar story about two spacetug captains.

Changing the point of view can alter the values we ascribe to certain measurements but cannot affect the reality which is being measured.