Turning Classic Physics Problems into Lab Practicums: Featuring the Patrolman and the Speeder

… encouraged by the testing of their solutions with the robots. They wanted to see it really work, and went back with enthusiasm to search for a more accurate method or to track down errors in mathematics.

Over beers at late night physics teacher gatherings, conferences like the American Association of Physics Teachers (AAPT) or Physics Teacher Camp, it seems someone almost always pipes up to say, “I just love lab practicums. They’re great! I wish I could do all lab practicums rather than textbook type problems.” Well, I and a collaborating teacher just turned some classic physics problems into lab practicums using my reprogrammed Scribbler 2 Robots. First is the patrolman and speeder problem, a great student exercise because it lends itself to so many different problem-solving approaches.

The Patrolman and the Speeder

Here’s the classic statement of the problem.

(a) A speeder driving down the road at a constant 20 m/s, passes a patrolman parked on the roadside. The patrolman waits 3 seconds, then pursues the speeder, accelerating at a constant 4.0 m/s². How long does it take the patrolman catch the speeder? How far has he traveled before doing so?

And this is all it takes to turn it into a lab practicum with the robots.

(b) This can be modeled with robots if we use cm instead of m and multiply the speeder’s velocity and the patrolman’s acceleration by 0.3 (i.e. if robot 1 starts with a speed of 6.0cm/s and robot 2 starts in pursuit 3 seconds later with an acceleration of 1.2cm/s²). Repeat your calculation with the numbers for the robots, then program and run them to test your conclusions.

I love this problem, because students can get the answer by solving two simultaneous equations leading to a quadratic formula, graphically through the intersection on a position vs. time graph, using graphical estimation based on area and the meaning of the slope on a velocity vs. time graph, or most elegantly as discovered by a group of my students in my first year of modeling by using a single line of simple algebra after setting up an equation using the velocity vs. time graph. Now they can also test their answers in the lab. (Details about how the students program the robot with graphical GUI’s are in a previous post.)

Typically student groups started by trying either the mathematically intensive simultaneous equations or else graphical estimation based on the the position or the velocity graph. The former groups would typically make mistakes – often leading to tests with the robots that were way off the mark – because this approach was stretching their mathematical background and ability, while the latter groups would more quickly arrive at an answer that was in the ballpark, but wonder how to get a better answer using a more deductive procedure.

By the time they were finished, most groups had done repeated tests and ended up using more than one method. This was definitely encouraged by the testing of their solutions with the robots. They wanted to see it really work, and went back with enthusiasm to search for a more accurate method or to track down errors in mathematics. The groups who started with the more mathematical approach would often wonder if there weren’t an easier way to get an accurate answer, and really appreciated the elegant mixed graph/equations approach based on the velocity vs. time graph. Observing the groups using graphical estimation helped them appreciate that approach as an aid to track down mistakes in their mathematics.

Students at this level are often uncomfortable with approximation and estimation as they transition from more concrete to more abstract reasoning. It’s not uncommon for students to undervalue graphical estimation or even feel personally diminished because they had to resort to that method. Both the method and the students using it got a boost when the robots confirmed that they were on track.

Crash!

Robot’s 1 and 2 start 131cm apart and move towards each other. Robot 1 travels with a constant speed of 15cm/s, while robot 2 starts from rest and has a constant acceleration of 3.0cm/s². Set the time in your programs so they stop simultaneously just before colliding.

Much as the last practicum, students were disappointed when the robots stopped short (which was fortunately more common than accidental collisions!), and wanted to improve their problem-solving method or experimental technique to get it closer.

This set of graphs brings up an issue, the distinction between going backwards and moving in the negative direction, that only seems to arise when applying these concepts in the lab. The graphs above programmed both robots with a forwards orientation, but the way they move towards each other, one of them should have a negative velocity. Similar issues have come up in past years when detecting motion of students or other objects with motion detectors, but never in a written problem with no lab component. As this confused some students, I’ll need to think more about how to address the issue next year.

Parking in the Garage

Google is developing these cool driverless robotic cars, which surprisingly few of my students had heard of. So what if you wanted to program one of these? Your speedometer reads 16m/s and your obstacle sensor detects your garage 160m away. How do you program your car to make a smooth stop in your garage? Well, just change m to cm and you can test it with our own robots. This a relatively easy, but fun lab practicum with a practical application that got students warmed up for the more difficult ones.

Did you notice that the robot backed out of the garage slightly? This was a great opportunity for a Socratic question to the students, who quickly figured out what they must have done ever so slightly wrong when they programmed it, thus previewing a new feature of the constant acceleration model.

A Prototype for a Subway Run

Engineers routinely build prototypes during the design process for a full scale system. I’ve given my students this problem for years, partly because it is so much easier to solve it graphically. This year, with just a few more minutes of work, they were able to test their solution with their own prototype. Although this problem is pretty finite, I like the concept of getting students to apply their physics to a more open-ended transportation design problem. The potential is there.

The classic problem:

A subway train starts from rest and accelerates at a rate of 1.8 m/s² for 15s. It runs at a constant speed for 50s and then slows down at a rate of 3.5m/s².

And this is all it takes to turn it into a lab practicum with the robots:

This can be modeled by a robot if we use cm instead of m and divide all the times by 3. (i.e. the robot would accelerate at 1.8cm/s² for 5s, run at constant speed for 16.67s (50s/3), and then slow down at 3.5cm/s²). Repeat your calculation with the numbers for the robot, then program and run it to test your conclusions. Programming a model to test a full scale system is common engineering practice.

As you can see, it travels very close to the predicted distance of 184cm. My goal at the beginning of the project was to program a robot so that its motion was accurate to within 1mm/s and 1mm/s². Over the course of 25s of motion, 1mm/s of error can add up to several cm, and the robots are at least that accurate – accurate enough and reliable enough that the students can tell when they’ve solved the problem correctly.

Differentiation

One advantage of being able to do so many different lab practicums with the same lab set-up was to allow students to choose their tasks. Following the garage practicum, some groups jumped immediately to the more difficult problems like the patrolman and speeder or car crash, while others preferred a more moderate-difficulty task like the subway problem or problem 4 below. The students were more engaged because they were given a choice, and I was easily able to move among groups checking work and asking probing questions.

4. Robot 1 travels with a constant velocity of 8cm/s for 18s. Robot 2 starts 6s after robot 1 with the same velocity of 8cm/s. Program robot 2 to catch robot 1 exactly at 18s by using just one constant acceleration.

Graph Matching again – Honors and AP Physics

My colleague, Jennifer Rotolo, who teachers the honors sections of physics and an APC mechanics class, had her students do an exercise based on the following velocity vs. time graph.

Here’s her description of the class.

Students were given a velocity vs. time graph along with a robot that was programmed to move according to that graph. Students were asked to describe exactly what the robot would do and then map out the robot’s motions on the floor. Students marked off how far forward they predicted the robot to move, how fast the robot would move, and where the robot would finish. Then they were able to watch the robot in action, to see how it actually moved. It was really neat to see all of the students’ eyes glued to that one robot! They were watching intently to see if their predictions were accurate.

The next task was for students to program their own robot to move exactly like the first robot. The catch was that students had to program their robots with position vs. time graphs, rather than the velocity vs. time graph they had just interpreted. Once students were finished, they watched their robot move alongside the first robot to see if the motions matched. The lesson was engaging for students and really helped them to grasp a tough concept that comes up in the first few weeks of physics. In the past, I have taught velocity vs. time graphs by talking through the qualitative motions out loud with students. The robots allow students to see these ideas for themselves, rather than me telling them.

The robots are a fantastic tool for learning physics. I think that one day every physics classroom will have them, and I feel very lucky to be one of the few that has them now. The robots can be used demonstrate endless concepts. They sort of make problems a game for students.

Here’s the graph-matching video.

The River

Toye Eskridge, our school’s marketing coordinator, described another of Jenny’s experiments for our school’s magazine.

In a lesson with her Advanced Mechanics class, Rotolo presented a classic physics problem that asks how long it takes a boat, moving perpendicular to a river current, to cross the river. “The idea is that the time it takes the boat to cross the river depends only on how fast the boat moves, and not on how fast the river flows,” she explained. “In other words, the time required for the boat to cross the river is independent of the current. This problem comes up for us in our study of vectors. We set up a lab in which some robots towed ‘rivers’ (poster board) to simulate a current, and some robots moved perpendicular to the current simulating the boats. The robots doing the towing were attached with string to the poster board. The students timed how long it took the ‘boats’ to cross the still river and then compared this time with how long it took the ‘boats’ to cross the river with current. They saw that the time was independent of the speed of the current. This is normally a tough concept to grasp, but I think seeing it with the robots really helped.”

And here’s what it looked like.

Conclusions

It was great fun creating these exercises and seeing some of the problems jump off of the printed page into real life. Because I could do so many things with just one apparatus that was intuitive for the students to use and did not require an elaborate set-up on my part, my class was even more hands on than usual. This is obviously just the tip of the iceberg.