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Dog Greets Owners

Some problems in HS physics omit detail then request an "answer" when only an "approximate answer" (as a "minimum" or "maximum") is obtainable. Students are expected to approximate to obtain an approximate answer. So, once solved it is proper to tell, as a consequence of approximation, is the answer magnitude smaller (or larger) than the truth and why. Consider the following physics text problem:

Statement:  "A dog runs back and forth between its two owners, who are walking toward one another. The dog starts running when the owners are 10 meters apart. If the dog runs with a speed of 3 m/s and the owners each walk with a speed of 1.3 m/s how far will the dog have traveled when the owners meet?"

i)  "A dog runs back and forth between its two owners..."   Students are likely to realize the system/event is three entities: a dog and two persons.

ii)  Some students are likely to approximate each entity (the dog and two persons) each as a BODY, each as a point moving in space. Others will imagine living persons, a crazy dog, all in motion. Better that the Author state something like "... approximate the dog and the persons as a BODIES."

iii)  "The dog starts running when the owners are 10 meters apart..." Students will know this initiates the event. "... when the owners meet." denotes the end of the event.

iv)  "If the dog runs with a speed of 3 m/s and the owners each walk with a speed of 1.3 m/s..."  The authors know it is not possible for a happy buoyant dog to move at 1.3 m/s for the duration of the event. What the author means is, "Assume the dog and persons move at constant speed. Thus the authors ask for an approximate solution.

v)  "... how far will the dog have traveled when the owners meet?"  The author means, "subject to the system models "BODY," and "assuming the turning of the dog requires no time" and "its speed is constant at 3 m/s..." Physics education is poor sometimes.

This Solution:   Let the persons be designated as "p1," "p2" and the dog as "d." The event is described in terms of the relative positions of the dog owners. We write that vector equation.

(1)
The position of person "1" plus
the position of "2 relative to 1"
equals the position of person "2."

Equation (1) specifies that the persons are ten meters apart at time equal to zero. The distance between them diminishes with time until it equals zero. Consequently the time the persons walk until they stand in the same space is:

(2)
To start the persons are 
10 meters apart. That 
distance diminishes 
2.6 meters each second.

So now we understand that the duration of the approximated event is 3.85 seconds. So what does the dog do during this time period. Knowing dogs, as we all do, we envision craziness... running, dancing, turning. It is difficult to proceed to solve this problem knowing dogs as we do.

The dog of the physics text author turns on a dime an always has constant speed: 3 m/s. The time that the author's dog can run is 3.85 seconds. Obviously the distance the author's (imaginary) dog can travel is:

(3)(3)

The message of this problem was the relevance of physics to dogs and owners. The premise is reasonable. The event, numbers and how it unfolds are believable. The author knew how to "find" (the word usually used) the missing fact, the "straight-line" distance traveled by the dog.

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