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Mysterious Benedict Society -- Salamander vs. Whisperer

Here, starting with a clear diagram, we can solve a "Salamander vs. Whisperer" problem. In the diagrams below,


The problem remains: Where to aim the transmitter? The procedure used to solve a problem like this may not be obvious, but it's not particularly complicated, either. (It might even be more educational than confusing.)

Step 1. Reflect the target (W) in the Reflective wall. Let W' (read "W prime") denote this virtual Whisperer.

Layout, W'

Step 2. Trace a path for the beam from the transmitter on the Salamander (S) to the target on the virtual Whisperer (W').

Step 3. Trace a path from the point where the beam hits the Reflective wall to the target on the real Whisperer.

Layout, W', Rays

Problem solved. Aim at vertical coordinate 2 on the Reflective wall.

The angle between the beam and the Reflective wall in this particular layout is about 33.7 degrees. (The exact value is the arctangent of 2/3. Because of the way that the positions of the objects were chosen, the beam travels two units horizontally for every three units it travels vertically. Knowing that, finding the angle itself is mere trigonometry. And, as we say in the trade, "The angle of incidence equals the angle of reflection.")

Layout, W', Rays, Angles

We might also observe that the angle between the incident beam and the reflected beam must be about 112.6 degrees (180 - 33.7 - 33.7 = 112.6), and thus there isn't a ninety-degree angle involved anywhere in the path of the beam.

And that's how a mirror works. An observer (S) sees a virtual image (W') of a real object (W). The virtual image (W') lies on a line passing through the object (W) and perpendicular to the mirror (R), equidistant from the mirror, on the opposite side. (All this applies only to a plane (flat) mirror. Curving a mirror adds complications, of course.)

As shown below, instead of reflecting the Whisperer, one could reflect the Salamander (S) in the Reflective wall, and then trace a path from the virtual Salamander (S') to the real Whisperer. (Here, a virtual observer (S') sees a real object (W).) That path crosses the Reflective wall at the same place, (0, 2), of course.

Layout, S', Rays

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© 2017 Steven M. Schweda.