Your basic motion sensor is Passive Infrared, basically it’s just looking at temperatures
They have a little IR sensor generally with a lens over it and the (fresnel) lens is designed so that something moving across it will cause abrupt jumps from high to low to high. The sensor picks up on these jumps and reports them as movement
Some use ultrasonic, basically sonar/echolocation where they send out a pulse of sound and listen to the echo. If it’s different than last time something has moved. These sensors can kind of see over/around barriers so they can be good for bathrooms
The general function behind most, whether it uses infrared, ultrasound, or even a webcam, is a game of “spot the difference.” When calibrated in a still room, the sensor knows that the general “image” it is “seeing” is a scene without any movement. When something changes, usually within a set tolerance level, (many are designed not to trigger the alarm when a pet or anything smaller than an average adult human wanders through the area) the sensor sees that there’s a significant difference in the “image” and hits the panic button. This can range from ultra-sophisticated 3D imaging using a complex network of sensors down to a beam of light going from full to dim/off. Even a tripwire rigged to a mousetrap does this on a mechanical level, though something like this is much easier to circumvent.
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