Velocity (and Speeding Tickets)

A standard physical interpretation of the ARC and IRC occurs when we consider motion.  We might measure the motion of a particle in a straight line.

As the position s of the particle changes over time, we can ask about the rate of change of position. This is velocity.

The average velocity of a particle is the ratio of distance traveled to elapsed time. For example, if a car travels 20 miles in one-half hour, then the average velocity of the car is

20 miles/0.5 hours = 40 miles per hour

In this situation, we view position (distance, displacement) as a function of time and the average velocity is again a slope, the ratio Delta s divided by Delta t.

The average velocity across the interval [t_1, t_2] can be expressed as

Average Velocity = (s_2-s_1)/(t_2-t_1)

where s is a function of time t.


The instantaneous velocity does not involve two moments in time, but one moment; it is the exact velocity of the particle at a given exact point in time and can be estimated by the average velocity. But we cannot easily compute the instantaneous velocity since the denominator of our ratio, the elapsed time, has shrunk to zero!

Average velocity and speeding tickets

When I was a teenager, driving around Macomb, Illinois in my red Volkswagen Beetle, I was told by friends about the speed trap out near the local park, Glenwood Park. According to my friends, a police car would sit off the road, in the bushes, near the top of a hill and catch speeders as the came over the hill.

The speed limit in that stretch of road was 35 miles per hour.

How did the police decide if I was speeding?

In those days, before radar, the police marked out a portion of the highway (let's say 100 yards or 300 feet) painting a line on the pavement at both ends of that stretch. Then when a car crossed one line, they would start a stopwatch. They would stop the stopwatch when the car crossed the second line. (Alternatives to painting a line on the road would be to note two clear landmarks, such as telephone poles, and measure the distance between them.)

For example, if the car took six seconds to cross the 300 feet, then the car was traveling at

300 ft/6 sec = 50 feet/sec 

This is just under 35 miles per hour so if one took 6 seconds to travel these 300 feet, they were good. No ticket@

The stopwatch in the police officer's hand is measuring time, which is in the denominator of our slope formula and so if the time was shorter, then one was obviously going faster.

Looking at this table, one can understand if the police officers decided to pull over anyone crossing the 300 feet in less than five seconds.  The severity of the ticket would depend upon whether one took 4.9 seconds or 4.0 seconds.  Someone who took 4.9 seconds has an average velocity just about 41 mph while someone who took 4.0 seconds had an average velocity above 51 mph.

The velocity here is an average.

One could crest the hill at 100 miles per hour (147 feet per second) but if they suddenly slammed on their brakes within the first second, they might be able to slow down enough to take 6 seconds to cross this interval and so appear to be within the law.

As a teenage driver, I knew that if I was driving too fast and if I suddenly saw a police car hiding in the bushes, I could quickly slow down and still be OK! One could drive fast if one were quick to spot hiding police cars!

How have police departments defeated this slam-on-the-brakes approach to speed limits? By finding ways to make the change in time much smaller and so make the average velocity much closer to the instantaneous velocity.

The modern radar gun (or the newer LIDAR speed gun) does exactly that. By measuring the doppler effect of a moving object on a radar pulse, the radar gun essentially picks up an instantaneous velocity, not an average velocity. If a car's instantaneous velocity has already been measured with a radar gun, then it does no good to slam on the brakes. (Indeed, if your velocity has already been measured by a radar gun, slamming on the brakes merely gives new, lower velocities, which make it clear that you knew you were traveling fast!)

Although the modern radar gun is much more accurate, the "stopwatch" method for measuring speeds is still use.  See this post of recent use in Pennsylvania.

Summary

We always prefer the IRC (or instantaneous velocity) to the ARC (or average velocity.) Technology in the middle of the twentieth century allowed police departments to measure average velocity but improving technologies eventually replaced the average velocity with the instantaneous velocity. (The police are happier... some of us not so much?)

We will look more at the IRC and velocity once we have an understanding of limits.