Experiment Problem in Kinematics: How Much Does it Take to Win the Race?

The goals for this activity are two-fold:

Content-related goals:
Students successfully completing this activity should demonstrate an understanding of

• position,
• time, and
• speed

as they apply to two objects moving simultaneously along the same axis.

Process goals:
Since this is an 'experiment problem' (as defined by Van Heuvelen et al) it encourages students to develop real-world problem solving skills such as

• adding definition to a problem
• planning a solution prior to diving into the problem
• dividing the problem into subproblems (fractionation)
• practicing good data-taking skills

Note that it is not realistic to expect students to develop these process skills from this activity alone - they will need repeated exposure to these processes.

Educational level: high school or introductory college
Setting: lab or lecture
Time required: varies, depending upon amount of provided information and setting - anywhere from 10 to 50 minutes (variations are described in the teaching tips below.)
Special equipment: 2 constant velocity objects, meter sticks, stopwatches, calculators
Pre-requisite knowledge: position, velocity, time, and the relationship between these quantities

(These instructions are written for an interactive lecture setting. Variations on this are discussed in more detail in the Teaching Notes and Tips that follow.)

1. Before class, measure the speeds of two constant speed objects (typically motorized cars - some examples are listed in the resources and references section below). Based upon this, choose a reasonable length of race for which one will beat the other by a minimum of a couple of seconds.
2. Still before class, if necessary, use tape or some other marker to indicate the starting and finishing lines for the race in the classroom. Run the race and determine the margin of victory.
3. In class, to start the activity, show the students the two objects moving.
4. Elicit from them that they both appear to move with constant velocity and that the velocities appear to be different. Sample guiding questions for this might include:
   What can you tell me about the motion of this object?
   How are the motions of the two objects the same?
   how are the motions of the two objects different?
5. Next, tell the students that you would like to race the objects against each other, but that before you actually have the race, you'd like them to tell you which one will win, and by how many seconds. Tell them to work in small groups to figure out what information they need to solve this problem. Some sample ways to phrase this question include:
   What information do we need to answer the question?
   What do we need to measure?
   What data do we need to take?
   Why would knowing that information be useful?
   
6. After they have had a few minutes to identify what they need to know, ask them to tell you what that needed information is. Tell them also that you may know some of the information already, but that they will have to figure out some of the information as a class.
   
   The information needed is:
   • The speed of the first object
   • The speed of the second object
   • The distance of the race.
7. Note that it takes a fair amount of time to measure the speed of both objects. In a lecture setting, you will probably want to provide them with the speed of at least one object, if not both. For information that you want them to determine via data-taking and analysis, some sample questions that you might use to assist them in verbalizing the process for you include:
   What equipment should you use to take the data?
   How many measurements should we take?
   (Measuring the distance for the race is a relatively quick measurement to have the students take in the class; ask for a couple of students to come up and execute the procedure under the guidance of the rest of the class.) A question that you may want to ask after the group has taken a measurement is
   Do you believe that measurement? Should you?
   For information that you are going to provide to them upon request, also ask them what process they would go through to determine it. Then when they've identified an appropriate method, say something like, "That's exactly what I did, and I came up with 0.30 m/s."
8. Once the class has the data they need, give them a few minutes to perform the calculations and determine their answers. Walk around the class and listen to the student conversations to get a sense of when you have given them enough time.
9. Poll the class for their answers. Ask a few representative groups to describe their process. If there are variations in procedure, have a discussion to determine whether the methods are equivalent or different, and if they are different, what the differences are. Give groups a chance to change their minds about the calculation method. Some sample questions here include:
   Did anyone else approach this differently?
   How many other groups used a similar strategy?
   Can someone from another group paraphrase what this group just said?
   Does this answer seem to be reasonable?
10. Run the race and have students take the data to see what the "right" answer is.
11. If some students did not arrive at the correct answer, revisit the calculations to help the students determine where errors were made.

There are a number of variations on this basic situation. Here are a few:

1. Rather than asking for the time elapsed between the two objects crossing the finish line, ask for the separation between the objects when the first one wins the race. This tends to be a little harder for students to measure when they run the race toward the end of the activity, but is not too hard.
2. When presenting the class with the initial problem, ask them, "which object will win the object, and by how much?" This leads to a discussion of what is meant by "how much," which could either be the amount of time or the distance between the objects when the first object crosses the finish line.
3. The instructor can choose to make students determine the speeds of one or both of the objects. This will make the activity run closer to the 50-minute mark.
4. To make the task more difficult, the question could be altered such that you want to find out how much of a head start (either time-wise or distance-wise) the slower object needs to have so that the two objects will cross the finish line at the same time.
5. A major variation on this is to set up the situation so that the two objects are moving toward each other and ask the class to determine where they will collide. (The students tend to enjoy seeing the collision.)
6. In a laboratory setting, each lab group could be asked to do the entire task as a graded activity, with the grade depending upon how close to the actual result their answer is.
7. Note also that this problem can be solved either mathematically or graphically.

First, it should be noted that this activity itself can be used as an assessment of whether students know how to analyze the motion of two constant velocity objects.

Second, options #1, 4, and 5 presented in the Teaching Notes and Tips above could work as assessments of this.

In a lab setting, each group of students can be assessed as described in #6 above. Note that if you do this and you are running several sections of lab that you may want to have a variety of race lengths and vehicle speeds available, so that you can create variations in the final answer.

In a lecture setting, the instructor can do some formative assessment of the class in general, depending upon what is said in the group discussions, as well as walking around and observing the class while they are discussing in their small groups.

A more formal assessment is possible by asking questions on quizzes, exams, or homework involving multiple objects moving at constant speeds. Some examples of questions like this can be found in Glencoe Physics: Principles and Problems (2005 edition), chapter 2, problems 57-59 (p. 53).

on student understanding of velocity
D.E. Trowbridge and L.C. McDermott, "Investigation of student understanding of the concept of velocity in one dimension," Am. J. Phys. 48, 1020-1028 (1980).

on experiment problems
A. Van Heuvelen, "Experiment problems for mechanics," The Physics Teacher 33, 276-280 (1999).
A. Van Heuvelen, L. D. Allen, and P. Mihas, "Experiment problems for electricity and magnetism," The Physics Teacher 37, 482-485 (1999).


Many instructors also refer to an exercise of this sort as a lab practicum.

examples of constant velocity vehicles
You can pick up a variety of battery-powered cars at a toy store that may work well for this. You could also order from a physics equipment supplier, such as one of the following:

Constant speed buggy from The Physics Toolbox (item M-09-F, http://www.vast.org/vip/VAST2010/ConstantSpeedBuggy/index.html)

Pasco variable speed motorized cart (item ME-9781, https://www.pasco.com/prodCatalog/ME/ME-9781_variable-speed-motorized-cart/index.cfm)