Dog Greets Owners

On vacation a woman took a morning run along a shoreline. While walking back she saw her husband and their dog walking toward her. Off-leash, the dog walked beside the man, sniffing and digging ... as dogs at the shore do.

At a distance from them the woman shouted, "DAISY Come!" Immediately the dog bolted full-speed toward the woman, ran around her, then ran full-speed back to the man, around him, then back to the woman... All the running of Daisy, their walking and laughing ended with the three together. The woman said, "Poor Daisy! She ran a lot further than we did."

The above is simply a description of an event. It does not constitute a "physical scenario" because no system was selected (and modelled), no quantitative information was available and no question (point of interest) was stated. There is only the woman's guess about about "How far the dog ran." Below we recast this event description to become a physical scenario. We select and model a system, supply further information, make reasonable assumptions then calculate the approximate distances involved.

Physical Scenario: Prior to the event the man and dog walk together in a straight line toward a woman who is walking toward them. We approximate their speeds to be constant at 2.5ft/sec. When the dog and woman are 180 feet apart, she shouts. Her shout (the event catalyst) marks time of event initiation. We estimate the speed of the dog to be 7.5ft/sec. As the man and woman move steadily toward eachother, the dog speeds back and forth between them. The even ends when the man and woman meet with the dog between them.

The scenario involves positions and velocities of the dog and persons in space and time. Such analyses are categorized as pure motion or kinematics. Two techniques we will review then use.

Vector Position ~ Single Entity

Relative Velocity ~ Two Entities

The space of the event are locations along a line. We take the line to be the "X" axis. Questions about the event are many, Here are a few:

What is the Duration of the event?
The motions of the man and woman can be used to answer this. Let's approach this with some generality.

Regarding motion of point "A" in the space OX with unit vector I we have:
The position of item "A" in OX-space over the time span from initial time t = t ^{+ until
time, t equals
(ii) the initial position is A plus the integral of the velocity of a over the}

(2)

To start the persons are 
180 feet apart. That distance diminishes 
5.0 feet each second.

What distance did the man walk and the woman walk?
What distance did the run?

When the event starts, the man and woman are 180 feet apart, moving toward eachother at 2.5 ft/sec.

Conditions and Assumptions:

The space of the motion (one-dimension) is along a straight line, we label as the X axis. The event is the sum of discrete increments of motion. Call them Stages.
Initial Positions and Speeds:

Man: Position, P_man = 0 ft I Constant Speed: V_man4 ft/s I

Woman: Position, P_woman = 360 ft I Constant Speed: V_woman4 ft/s (-I)

commences when the dog hears, "Daisy come!"upon the is scenario description is reasonably complete but it lacks notations and specifics. Initial conditions are: man (and dog) a distance from the woman moving at constant speed V_walk, each toward the other. The event commences upon the woman's yell. The man and woman proceed steadily in linear space and time - along a line, toward each other. This is clear, needs no approximation.

The motion of the dog is less easy to prescribe. erratic. The dog accelerates to V_d, moves to the .

One style of academic texts and general thinking about physical system/events is an approach wherein one knows (or had been told) facts about a system/event, those facts thererafter to be used to deduce or calculate some other fact being consitent with approximations aand laws of physics.

the generic idea in education parlance is "given these things known" therafter "solve" (in accord with the laws of physics) to determine something else, the "answer" of its query. It is easier to do this when a complete, through specification of system/event/charistics are known.

=== OLD ======

Virtually all text-problem-statements 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 72 feet 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 7.5 ft/s and the owners each walk with a speed of 2.5 ft/s..." The authors know it is not possible for a happy buoyant dog to move at 7.5 ft/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 
72 feet apart. That distance diminishes 
5.0 feet 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.