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 to her then 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 paragraphs constitute a description of a physical event. Notice no system was specified, no quantitative information nor question (object of inquiry) was stated. There is only the woman's conjecture about "How far the dog ran compared to them."

Below we recast this event description to become a physical scenario. We will select a system and system model. For other information needed we will 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 the time of event initiation. We designate the location of tghe man and dog as "0 ft." on an X-axis. 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 event 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. Position and velocity are vector quantities. To specify a vector requires specification of a vector space and a unit-vector basis.

Vector Space and Basis: Our vector space is a straight line that extends from the man and dog to the woman. We label this the X-axis. At the start (the moment the woman shouts)

the dog and man are located at X = 0 ft. Each has the position vector: P = 0 I.

Also the woman is located at X = 180 ft with position vector P_woman = 180IIWe define the unit vector 180 ft on the X-axis." Atch of the initial conditions is:

(2)

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

   |
---|-------------

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.