## It's a Trap!

Imagine you're late to an important appointment. You're driving on the highway in a 60 mph zone, but you look at the clock and realize that at that rate, you'll never get where you're going in time. Throwing caution to the wind, you press your foot against the accelerator and move into the fast lane. You relax a little bit as you pass other cars, but then you hear the cry of a police siren behind you, and reality sinks in.

Hopefully you've never found yourself in this situation. But if you have (or if you've seen it happen on TV), then you know that the police are able to tell you with a fair degree of precision (within 1 mph or so) how fast you were traveling. Obviously, they aren't estimating your speed as they watch you shoot down the highway; instead, officers have a variety of technologies to help measure the speed of cars on the roads. One of these technologies is called LIDAR (a mutt of a word if ever there was one, being a portmanteau of "light" and "radar," the latter of which is itself an acronym). LIDAR uses laser beams to track the speed of moving objects.  But how exactly does this technology work, and can it be trusted?

The basic premise is this: when an officer wants to track the speed of your car, he or she fires a beam from the LIDAR gun.  The beam hits your car and reflects back to the gun; since the speed of light in air is a known quantity, the time it takes for the beam to return tells the gun how far away your car is.  By sending out a number of these beams, the gun can track your position over time, and from that it can deduce your speed.

For example, suppose the gun is in front of your car as in the diagram above.  It fires a beam that returns one microsecond (1 × 10-6 s) later.  Since the speed of light is around 9.84 × 108 ft/s, this means that the distance traveled by the light beam is equal to 9.84 × 108 × 1 × 10-6, or 984 feet.  Since this covers both the trip to the car and the trip back, the car must be 984 ÷ 2 = 492 feet away.

Next, imagine that the gun takes another reading 0.4 seconds later, and finds that the light takes 0.91 microseconds to get to the car and back.  This means that the car is now just 447.72 feet away, meaning that it has traveled 44.28 feet in 0.4 seconds.  This puts its speed at around 110.7 feet per second, or roughly 75 miles per hour.

Cool, right? There's just one thing. We assumed here that the gun is in the path of the car's motion, which probably isn't true. Cops don't stand in the road to look for speeding cars; usually they're off to the shoulder. And this has an interesting effect on the LIDAR gun's measurements. For example, let's imagine the same scenario, but with the gun 150 feet away from the road:

Now the first distance measurement the gun makes (the green line) is the hypotenuse of a right triangle with side lengths of 150 feet and 492 feet.  Therefore, the square of this length is equal to 1502 + 4922 = 264,564 ft2, and so the length is equal to the square root of this value, or around 514.36 feet.  Similarly, the second measurement the gun makes (the red line) will now have a length of around 472.18 feet.  Notice that even though the gun measures longer distances, the difference between them (42.18 feet) is now shorter than before, meaning that the gun will record a lower speed (roughly 72 mph)!

In fact, you can use a little bit of geometric ninjitsu (in the form of the triangle inequality) to show that this is always the case: assuming the gun is calibrated correctly, whenever it's off to the shoulder as in the diagram above it will always record a speed that is less than the true speed.  So the next time you (or someone you love) gets pulled over for speeding, you may want to think twice about contesting the ticket.

Teachers: want to have this conversation with your students? Then check out our newest lesson, It's a Trap!

## 2 thoughts on “It's a Trap!”

1. Sammie says:

I love this lesson! I discovered a typo, though. Your student handout, on question number 5 says the officer stands 50 feet from the side of the road, but the lesson guide says 150 feet, and gives an answer based on 150 feet.

1. matt says:

Good catch, Sammmie, thanks! We've fixed the typo.