Adventures in Radar Detection

As I was driving down the highway the week before last, I got to thinking about all the police cars and their radar speed detectors I was passing. Thankfully I am not a big speeder and did not have to learn from personal experience the effectiveness of their radar guns. But it did make me curious how exactly the radar gun was used to determine a car's speed.

I knew it had to be using the Doppler Effect somehow. You know when you are driving along and you hear a fire engine or police siren behind you, and how the pitch shifts as the emergency vehicle passes you? That's the Doppler effect on sound waves.

What happens is, the wavelength of the sound gets compressed as the siren approaches you. Conceptually, the siren is producing a sound wave that peaks every so often (hundreds or thousands of times each second). In the case of a stationary siren, these wavefronts would be equally spaced, but in the case of a moving siren, the wavefronts are going to be squished together in the direction the siren is going, and spaced farther apart in the opposite direction.

Mathematically speaking, we know the wave equation: v = f, where v represents the velocity of the wave,  is the wavelength, and f is the frequency. Let's suppose, to keep things simple, that you (the observer) are stationary, and the siren is moving. The frequency of the wave that you observe is proportional to the velocity of the observed wave divided by the frequency of the observed wave (i.e., f_o = v_o/_o). The velocity of the wave is constant; the siren isn't pushing the air faster, it's altering the location and frequency of the wavefronts. So v_o = v (a constant). How does the movement of the siren change the wavelength of the wave? Well, there is some amount by which the wavelength _o gets changed from the at-rest wavelength ; let's call that a. So _o = -a, making our equation
f_o = v/(-a).

The shift in wavelength, a, has to be proportional to the velocity of the siren, v_s (i.e., v_s = af). We can substitute v/f for , and v_s/f for a, by solving for the wavelength in the wave equation. Thus we obtain
f_o = v/(v/f - v_s/f)
f_o = (v/(v-v_s)) f.

We can go through this exercise again, with the siren stationary and the observer moving, and obtain the equation
f_o = ((v-v_o)/v) f,
and then combine the two equations into the more general
f_o = (v-v_o)/(v-v_s) f.

Of course, a radar gun uses electromagnetic waves, not sound waves. When v is much, much bigger than v_o or v_s, then we can simplify this equation somewhat. Suppose we multiply the right-hand side, ((v-v_o)/(v-v_s)) f, by (v+v_s)/(v+v_s), or in other words, by 1. Then we obtain
f_o = f (v-v_o)(v+v_s)/((v-v_s)(v+v_s))
= f (v² - (v_s-v_o)v - v_ov_s)/(v²-v_s²).

This doesn't look very nice or helpful, but since the velocities of the source and the observer are so tiny compared to the velocity of the wave, then we can cancel out any second-order (i.e., squared) terms in v_o and v_s:
f_o = f (v² - (v_s-v_o)v - v_ov_s)/(v²-v_s²) = (v² - (v_s-v_o)v)/v²

So finally, we end up with the much simpler equation
f_o = f (1-(v_s-v_o)/v).

This is the equation that is used in a radar gun. The radar gun shoots out some radio waves, which bounce off your speeding car. The frequency shift is used to determine the speed of your vehicle.

It is slightly more complicated, however, when faced with identifying the actual culprit who is speeding, as well as correcting for certain types of errors that crop up when the radar is not trained in the exact direction of the car's velocity, etc. But combining the radar gun readings with other techniques, such as observation, speed matching, etc., police are able to catch speeders with better and better accuracy.

Sources:
How Radar Guns Work
Doppler Effect (Wikipedia)
Doppler Shift (Eric Weisstein's World of Physics)

More on Cars

I am extremely obsessed with figuring out how to get the best gas mileage out of the car I drive, for a variety of reasons.

First, I think about my contributions to global warming every single day as I make my trip to work. (If only they would make a solar-powered car capable of driving at least 15 miles on a trip, I would be all set.) Second, I am an amateur physicist and I want to understand the workings of the vehicle in which I spend more than five hours a week.

But most relevantly, I am a mathematician, and the idea of optimizing gas mileage as opposed to other possible variables in the equation of driving (such as time or number of miles traveled) sounds really fun and challenging to me, because in the car I drive, I can only ascertain this indirectly.

The car has a manual transmission, so I can control which gear it is in. It also has a crude tachometer, which I can read only to the nearest 125 RPM, a speedometer which I can read to the nearest half-mile per hour, and an odometer which measures distance traveled to the nearest tenth of a mile. Can I use only these tools to optimize my gas mileage on my daily commute?

This is what I've been trying to figure out, actually. I wish I had a good answer. The problem is that I am currently lacking a good intuitive understanding of the way the car works. I've been reading up on torque, horsepower, and RPM, and trying to put it all together. I'll let you know when I finally get there.

The Evolution of Watches

I saw this video, via Greta Christina, and I thought it was so cool that I wanted to share it with you, my vast blog audience.

Suppose that gears, springs, and hands had properties that attracted them to one another like biomolecules do. What would happen if "creatures" consisting of these building blocks of "life" were allowed to mate and mutate, with the "environmental pressure" that the ones best able to tell time would survive to pass along their "genes"?

They would create clocks, that's what would happen!



The author of this movie wrote some code to look at how this would happen. I'm looking forward to downloading his code and checking it out.

Adventures in Measurement

It's often important to quantify the properties of objects around us. For example, if we were selling apples, we would need to have a way to quantify them in order to set consistent prices. We might decide to quantify the apples by number, weight, or volume. Maybe we'll sell 3 for $1, or 50¢ per pound, or $5.00 per peck.

The study of the science of measuring is called metrology. Metrologists concern themselves with several important questions: How can we quantify the properties of objects? What units of quantity are meaningful, and how do we measure them? How do we apply these methods of measuring in the real world? How can we create measurement systems and apply them in a manner that helps regulate trade, taxation, safety, etc.?

How do we assure that measures are accurate? For example, if I'm pricing apples by weight, how can my customers be assured that they're getting a fair deal?

The scale I use to measure the weight of the apples is calibrated to a certain accuracy, which assures that the "pound" of apples I sell you isn't lighter than the "pound" of apples somebody else sells you, at least within a certain margin of error. The pound is officially defined as 0.45359237 kilograms, and the kilogram is officially defined in relation to an artifact, the International Prototype Kilogram (IPK). "The IPK is the kilogram... [it] is made of a platinum-iridium alloy and is stored in a vault at the BIPM in Sèvres, France." The United States owns three replicas of the IPK, housed at NIST (National Institute of Standards and Technology).

How accurate is the scale I use to measure apples? There are NIST standards which the scale would have to meet before I could use it to sell apples. At a minimum, it would have to be accurate enough that the uncertainty of the measurement (e.g., x lbs ± y lbs) would not impact the cost of the product. So, for an uncertainty of y when buying x pounds of apples, the cost of x+y lbs should be the same as the cost of x-y lbs.

How could that be, you may ask, since x+y ≠ x-y (unless y=0)? Well, when we deal with money, we don't pay fractions of a penny; merchants round to the nearest penny. In other words, if I charge 50¢/lb, then if I weigh out an amount of apples a that has a "true" cost c such that 49.5 ≤ c < 50.5. If c is defined by the previous inequality, and c = 50 a, then what are the upper and lower limits on a?

Well, if c = 49.5, then 50 a = 49.5, and therefore a = 0.99. At the other extreme, if c = 50.5, then 50 a = 50.5, and therefore a= 1.01. Thus my scale would need to be calibrated such that it measures one pound to within an accuracy of 0.01 pounds. If we convert this to relative error, its percent error must not exceed 1%.

For measuring apples, we would need a much higher accuracy than we would need for measuring the weight of trucks at a highway weigh station. We don't need to know the weight of the truck to within a hundredth of a pound, but we might still need the same relative error (i.e., 1%).

Also, sometimes we don't need to know the exact size of something, we just need a ballpark figure. For example, if we're trying to fit a couch into the back of a pickup truck, we just need to make sure that the couch is shorter than the length of the truck bed.

Other times, we want the most accurate measure we can get, but are limited by uncertainty or human error. For example, if we want to measure the length of an ant, but all we have to do it with is a yardstick, we will be limited by the size of the calibrations on the yardstick, and by the ability of our eyes to see something so small. If we had better equipment (i.e. a smaller, more finely calibrated ruler, and a magnifying glass) we would get a more accurate measurement.

One of the things that makes me roll my eyes every time I watch American football is the questionable way in which the status of the downs is sometimes determined. The referees have a chain of length ten yards, anchored by bright orange poles at each end, which is handled by the chain crew. The chain crew holds one end of the chain on the sideline at a point that is parallel to the location where the ball starts upon first down. The length of the chain is stretched along the sideline in the direction the team is driving, to determine whether the team has made a first down. Sometimes, when it is a close call, the chain crew moves the chain onto the field, and the referees measure with it, to see if the first down has been achieved. Sometimes, by a matter of inches, the team doesn't make the down.

There are some problems with the manner in which the status of the down is determined. We need to see the errors inherent in the system, which compound to make this measurement wrong.

First, the starting point at which the chain is placed is lined up visually with the location of the ball more than twenty-five yards away (In the NFL, the field is 160 feet wide). If the angle between the ball, the end of the chain, and the sideline is off by one tenth of a degree (i.e. it's 89.9° or 91.1°), then the difference between the starting point of the chain and the true starting point of the ball is off by more than 1.5 inches.

Second, when the down attempt is over, the referee generally places the ball in the place he believes is the point of farthest advance. How accurate is his placement? This is something I don't know, but it can't be more accurate than within a few inches. Let's say within three inches for the sake of argument.

Let's assume that the length of the chain is perfectly accurate. (It is probably off by some small factor; also, we are neglecting shrinkage or expansion due to temperature, but that's okay.) Even so, in the worst possible case, the starting point of the chain was 1.5 inches ahead of the starting point of the ball, and the referee put the ball down three inches behind the line of farthest advance, meaning that the difference between the true advancement and the measured advancement exceeded 4.5 inches. Given that a football is about 11 inches in length, that means the measurement could be off by more than 2/5ths of a football length! I've seen downs decided by less than that!

So what happens in football is that they think they have an accurate measurement of the lengths involved, but in reality, the fate of a team's advancement down the field is determined by little more than luck.

This is a real-life example of an organization needing a consultation with metrologists. If I were the football commissioner, that would be one of the first things I did.

Metrology is a fascinating subject. It's about more than weights and lengths; metrologists also determine how to measure volume, time, energy, work, and many more qualities. If I weren't a computational scientist, I think I'd want to be a metrologist. I hope you enjoyed this foray into metrology as much as I did!

Adventures in Applied Physics

My favorite part of my undergraduate major was studying kinematics. My favorite fundamental force is gravity. Unfortunately, there's not much more to discover in the field of kinematics, so I had to find something else to study in grad school.

But I am still an amateur kinematicist, and I perform experiments in the field nearly every day. I have a ten-mile commute to work every day (20 miles round trip), and it can get boring, so I amuse myself by attempting to leverage the force of gravity, friction, and my own ingenuity to use as little fuel as possible. I don't drive a Prius, but I strive for Prius-like mileage in my 2005 VW Beetle. It is a stick shift so I have complete control over what gear it's in. I'm also very familiar with the route, so I know where the hills and stoplights are.

I have two heuristics that I employ to minimize gas consumption: brake as little as possible, and coast as much as possible. These can be conflicting objectives, however, because when you're going down a hill towards a red light, you need to brake rather than take advantage of gravity, the natural accelerator. So there's a balance.

Also, braking is sometimes necessary, to avoid speeding and also to prevent stopping. Stopping should be avoided, because accelerating from 5 mph consumes 20% less fuel than accelerating from a full stop. Braking can be reduced by anticipating the behavior of the drivers around you. I stay way behind the car in front of me, so that I have the time to make fuel-efficient adjustments (e.g., coasting uphill, light braking) to my speed.

Because I am super geeky and I have my art merit badge (see the previous post), I decided to draw for you, my vast blog audience, a crude map of my daily journey. It's crude because first of all I freehanded it without looking at a map, and second of all I added my own contours that are probably completely inaccurate with respect to the true elevations, but they are my impression of the lay of the land. Finally, it is totally not to scale. (Part D of the map is more than half the journey, but it's boring so it doesn't deserve much space.)



The green circle represents home, and the red stop-sign represents work. The thick black line is the route I take, the thinner black lines are roads making up important intersections with stoplights, and the rainbow lines represent the elevation contours, with red being the highest elevation and descending through orange, yellow, green, and blue. So, work is at a lower elevation than home, meaning that my journey to work is mostly downhill, while my journey home is mostly uphill. But, you can see that there are some major hills in between: at A there is a valley, and at C there is a major hill. B is through the city, where it is mostly level but there are a number of stoplights. Somewhere along D, there is a guard shack, where you have to stop and show your badge in order to get on company property.

I can usually coast all the way from the intersection of my street with the main road (at the red contour) to the second intersection because that first stoplight is usually green. If I'm super lucky I can make it through the second stoplight too, although that usually doesn't happen. If I see it's red I just take the car out of gear and gently brake, in hopes that I won't have to stop if I arrive at the intersection later. (Of course I stop if I get there and the light is red.)

Part B is the city and if I go at just the right speed I can hit all the green lights. It is basically flat and pretty dull, except that there is this one intersection where the brains of people turning left into the Kroger shopping center atrophy or something. A statistically significant percentage of the times I am stopped at that light, the people in the first car in the line turning left into Kroger don't notice that they can go until it's almost too late. I don't know what it is: the radiation, the electromagnetic fields, or what, but this happens at least once a week.

It's not until I get to Part C of my jouney that things become kinematically interesting once again. There is a huge hill. I start at the bottom of the hill at 45 mph (the speed limit) and if nobody is behind me I take the car out of gear near the crest. At the top of the hill I am down to 25 mph, but once gravity (the natural accelerator) takes over, I begin to speed up, ending up at 45 mph at the bottom of the hill. Usually I can merge into traffic on the main road of part D without slowing down too much.

As I approach the gate, I take the car out of gear and coast. This probably drives the people behind me crazy, but relying on friction to decelerate saves a lot of fuel. I do usually have to brake, because I haven't figured out where exactly I should start coasting.

The 2005 VW Beetle gets 25 mpg city, 30 mpg highway, but I get better mileage than that. I estimate that I get about 33-35 mpg, which is over 20% more efficient than the standard.

New Car

I'd been looking for a new (to me) car, because Gundar the 1990 Volvo has become a petulant teenager and once again believes that he'll start when he dang well feels like it rather than every time we want him to. Instead of trying to get him repaired, we decided to just get another car.

It didn't need to be a very big or luxurious car like the new Impala we have for the babymobile; it just needed to get me from point A to point B and back (i.e. home to work). I decided that I didn't need to have a brand new car, but it would be strategic to get a car three years old or less, because it could still have some warranty left.

We found a good car on the internet: a 2005 Beetle with less than 5000 miles, at a dealership. On Sunday we went to testdrive it. It was really nice: a 5-speed stick shift, silver (not my favorite color; silver is just "shiny gray" and I hate gray), surprisingly roomy for such a small car. Even Jeff could fit in it comfortably.

So on Monday we went to our bank and got a car loan (a blank check, preapproved up to a certain amount). Then we went to the dealership and bought the car. We parked it near the dealership before going on to the hospital to see Uncle Wayne (who was finally in a recovery room and got to meet Vinny). After our visit, I drove the new car home while Jeff drove the other car.

I had missed driving a stick shift. With all the hills here, it is much more fun to drive than it was to drive Ingrid the 1982 Volvo around flat Illinois. I am so looking forward to coasting down hills and enjoying physics in action!